algebra.bigb

= Algebra
{wiki}

= Field
{parent=Algebra}
{wiki=Field_(mathematics)}

A field is a commutative ring in which every nonzero element has a multiplicative inverse.

= Algebraically closed field
{parent=Field}
{wiki}

A field is algebraically closed when every nonconstant <polynomial> over it has a root in it, equivalently when every such polynomial splits into linear factors.

= Characteristic of a field
{title2=$\operatorname{char}F$}
{parent=Field}
{wiki=Characteristic_(algebra)}

The characteristic of a <field> $F$ is the least positive integer $n$ for which $n\cdot1_F=0$, or zero if no such integer exists.

= Characteristic
{synonym}

= Polynomial
{parent=Algebra}
{wiki}

A polynomial is a finite sum of monomials with coefficients in a ring.

= Monic polynomial
{parent=Polynomial}
{wiki}

A monic polynomial has leading coefficient one.

= Root of a polynomial
{parent=Polynomial}
{wiki=Zero_of_a_function}

A root of a polynomial $f$ is a scalar $a$ satisfying $f(a)=0$.

= Roots of a polynomial
{synonym}

= Multiple root
{parent=Root of a polynomial}
{wiki=Multiplicity_(mathematics)#Multiplicity_of_a_root_of_a_polynomial}

A root $a$ has multiplicity $m$ when $(X-a)^m$ divides the polynomial but $(X-a)^{m+1}$ does not. It is a multiple root when $m>1$.

= Common root
{parent=Root of a polynomial}

A common root of polynomials $f$ and $g$ is a scalar $a$ satisfying $f(a)=g(a)=0$.

= Elementary symmetric polynomial
{parent=Polynomial}
{wiki=Elementary_symmetric_polynomial}

The elementary symmetric polynomials in $x_1,\ldots,x_n$ are the sums of all products of $k$ distinct variables, for $k=1,\ldots,n$.

= Fundamental theorem of symmetric polynomials
{c}
{parent=Elementary symmetric polynomial}
{wiki}

Every symmetric polynomial over a commutative ring has a unique expression as a polynomial in the elementary symmetric polynomials.

= Newton identities
{parent=Elementary symmetric polynomial}
{c}
{wiki=Newton%27s_identities}

Newton's identities relate the elementary symmetric polynomials $e_k$ to the power sums $p_k=\sum_i x_i^k$. The first three give
$$
e_1=p_1,
\qquad
2e_2=p_1^2-p_2,
\qquad
6e_3=p_1^3-3p_1p_2+2p_3.
$$

= Irreducible polynomial
{parent=Polynomial}
{wiki}

An irreducible polynomial is a nonconstant polynomial that cannot be written as a product of two nonconstant polynomials over its coefficient field. Over a general integral domain, neither factor may be a unit.

= Polynomial division
{parent=Polynomial}
{wiki=Polynomial_long_division}

For polynomials $q$ and nonzero $Q$ over a field, there are unique polynomials $s,r$ such that $q=Qs+r$ and $\deg r<\deg Q$.

= Degree of a polynomial
{title2=$\deg$}
{parent=Polynomial}
{wiki=Degree_of_a_polynomial}

The degree of a nonzero polynomial is the largest exponent of a monomial having nonzero coefficient.

= Lagrange root bound over a field
{parent=Degree of a polynomial}
{c}

A nonzero <polynomial> of <degree of a polynomial>[degree] $d$ over a <field> has at most $d$ distinct roots. Indeed, each root supplies a linear factor by <polynomial division>, and induction on $d$ gives the bound.

= Quadratic function
{parent=Polynomial}
{wiki=Quadratic_function}

A quadratic function of one variable has the form $ax^2+bx+c$ with $a\ne0$.

= Quadratic equation
{parent=Quadratic function}
{wiki}

The roots of $ax^2+bx+c=0$, where $a\ne0$, are
$$
x=\frac{-b\mathbin\pm\sqrt{b^2-4ac}}{2a}.
$$

= Algebraic number theory
{parent=Algebra}
{wiki}

= Algebraic integer
{parent=Algebraic number theory}
{wiki}

An algebraic number is an algebraic integer when it satisfies a monic polynomial with integer coefficients. Algebraic integers are closed under addition and multiplication.

= Ring of integers of a quadratic field
{parent=Algebraic number theory}
{wiki=Ring_of_integers}

For square-free $d\ne0,1$, an integral basis is $\{1,(1+\sqrt d)/2\}$ when $d\equiv1\pmod4$, and $\{1,\sqrt d\}$ otherwise. The discriminant is respectively $d$ and $4d$.

= Splitting of rational primes in a quadratic field
{parent=Ring of integers of a quadratic field}
{wiki=Splitting_of_prime_ideals_in_Galois_extensions}

For a quadratic field of discriminant $D$ and a rational prime $p$, the ideal $(p)$ splits into two distinct prime ideals, remains prime, or ramifies according as the quadratic character $\chi_D(p)$ is $1$, $-1$, or $0$. The corresponding prime-ideal norms are $(p,p)$, $p^2$, and $p$, respectively.

= Ramification of a prime
{parent=Splitting of rational primes in a quadratic field}
{wiki=Ramification_(mathematics)}

A prime $p$ ramifies in a number field when its prime-ideal factorization contains a factor with exponent greater than one.

= Splitting of three in Q of square root minus p
{parent=Splitting of rational primes in a quadratic field}

If $p\equiv11\pmod{12}$ and $K=\mathbb Q(\sqrt{-p})$, then $\mathcal O_K=\mathbb Z[(1+\sqrt{-p})/2]$. The generator's minimal polynomial reduces modulo $3$ to $T(T-1)$, so $(3)$ splits into two distinct prime ideals of norm three.

= Discriminant of a biquadratic field
{parent=Algebraic number theory}

The discriminant of a biquadratic field is the product of the fundamental discriminants of its three quadratic subfields.

= Algebraic integer module criterion
{parent=Algebraic number theory}
{wiki=Integral_element\#Equivalent_definitions}

An element $x$ is integral over $\mathbb Z$ exactly when it acts stably on some faithful nonzero finitely generated $\mathbb Z$-module containing one.

= Ring generated by algebraic integers
{parent=Algebraic integer module criterion}

A ring generated over $\mathbb Z$ by finitely many algebraic integers is a finitely generated $\mathbb Z$-module.

= Integral dependence is transitive
{parent=Algebraic integer module criterion}
{wiki=Integral_element\#Transitivity}

If $B$ is integral over $A$ and an element is integral over $B$, then it is integral over $A$.

= Integral basis
{parent=Algebraic number theory}
{wiki}

An integral basis is a $\mathbb Z$-basis of the ring of integers of a number field.

= Field discriminant
{title2=$d_K$}
{parent=Integral basis}
{wiki}

The field discriminant is the discriminant of any integral basis and controls the index of every suborder.

= Discriminant-index formula for an integral lattice
{parent=Field discriminant}

Let $\omega_1,\ldots,\omega_n$ be an integral basis and let $\alpha_j=\sum_i a_{ij}\omega_i$ be a rational basis of algebraic integers, with integer matrix $A=(a_{ij})$. Their trace Gram matrices satisfy
$$
G_\alpha=A^TG_\omega A,
$$
so
$$
\Delta(\alpha_1,\ldots,\alpha_n)
=(\det A)^2d_K.
$$
The absolute discriminant is minimized exactly when $A$ is unimodular, equivalently when the $\alpha_i$ form an integral basis.

= Cubic field of discriminant minus 307
{title2=$K=\mathbb Q(\alpha),\ \alpha^3-5\alpha+8=0$}
{parent=Algebraic number theory}

The polynomial $X^3-5X+8$ is irreducible over $\mathbb Q$. For
$$
\beta=\frac{\alpha+\alpha^2}{2},
$$
one has
$$
\beta^3-5\beta^2+11\beta-12=0,
\qquad
\alpha=-4+3\beta-\beta^2.
$$
The field has ring of integers $\mathbb Z[\beta]$ and field discriminant $-307$.

= Integral basis of the cubic field of discriminant minus 307
{parent=Cubic field of discriminant minus 307}

The elements
$$
1,\qquad\alpha,\qquad\beta=\frac{\alpha+\alpha^2}{2}
$$
form an integral basis. Relative to this basis, multiplication by $\beta$ has integer matrix
$$
\begin{pmatrix}
0&-4&-4\\
0&2&-1\\
1&1&3
\end{pmatrix}.
$$
The basis discriminant is one quarter of the discriminant $-4\cdot307$ of $(1,\alpha,\alpha^2)$, hence is the square-free integer $-307$.

= Prime ideals above two and three in the cubic field of discriminant minus 307
{parent=Cubic field of discriminant minus 307}

For $g(X)=X^3-5X^2+11X-12$, the reductions are
$$
\overline g=X(X^2+X-1)\quad\text{over }\mathbb F_3
$$
and
$$
\overline g=X(X^2+X+1)\quad\text{over }\mathbb F_2.
$$
Both quadratic factors are irreducible, so
$$
(3)=(3,\beta)(3,\beta^2+\beta-1)
$$
and
$$
(2)=(2,\beta)(2,\beta^2+\beta+1).
$$

= Field norm
{parent=Algebraic number theory}
{wiki}

The field norm $N_{L/K}(x)$ is the determinant of multiplication by $x$, equivalently the product of its conjugates when the extension is separable.

= Norm obstruction to being a square
{parent=Field norm}

If $x=y^2$ in a number field, then $N(x)=N(y)^2$; a norm that is not a square in the base field therefore obstructs square roots.

= Unit from norm one or minus one
{parent=Field norm}

An algebraic integer whose absolute norm is $1$ or $-1$ is a unit, because its conjugate product supplies an integral inverse.

= Minkowski convex body theorem
{c}
{parent=Algebraic number theory}
{wiki=Minkowski%27s_theorem}

If $\Lambda\subset\mathbb R^n$ is a full lattice and a measurable convex centrally symmetric set $C$ has volume greater than $2^n\operatorname{covol}(\Lambda)$, then $C$ contains a nonzero lattice point.

= Minkowski embedding of a real quadratic field
{parent=Minkowski convex body theorem}

For a real quadratic field $K$ with embeddings $\sigma_1,\sigma_2$, the image
$$
\{(\sigma_1(\alpha),\sigma_2(\alpha)):\alpha\in\mathcal O_K\}
$$
is a lattice in $\mathbb R^2$ of covolume $\sqrt{|d_K|}$, and the product of the two coordinates is the field norm.

= Mahler measure
{parent=Algebraic number theory}
{wiki}

If $f(X)=a_d\prod_{j=1}^d(X-\alpha_j)$, its Mahler measure is
$$
M(f)=|a_d|\prod_{j=1}^d\max\{1,|\alpha_j|\}.
$$

= Mahler-measure norm argument at measure two
{parent=Mahler measure}

Let an irreducible $f\in\mathbb Z[X]$ have $M(f)=2$ and a root outside the closed unit disk. Its leading coefficient must have absolute value one. The product $B$ of all roots outside the unit disk is then an algebraic integer with $|B|=2$; conjugate pairs make $B$ real, so $B=\pm2$. The quotient of the norm of any root by $B$ is both rational and an algebraic integer, and its absolute value is at most one. It is therefore $\pm1$, so every root has absolute norm two.

= Dirichlet unit theorem
{parent=Algebraic number theory}
{c}
{wiki=Dirichlet%27s_unit_theorem}

If a number field has $r$ real embeddings and $s$ conjugate pairs of complex embeddings, then
$$
\mathcal O_K^\times\cong\mu(K)\times\mathbb Z^{r+s-1},
$$
where $\mu(K)$ is the finite group of roots of unity in $K$.

= Units of the quadratic field Q square root of five
{parent=Dirichlet unit theorem}

For $K=\mathbb Q(\sqrt5)$,
$$
\mathcal O_K=\mathbb Z\left[\frac{1+\sqrt5}{2}\right].
$$
Writing $\phi=(1+\sqrt5)/2$, the full unit group is
$$
\mathcal O_K^\times=\{\mathord\pm\phi^n:n\in\mathbb Z\}.
$$

= Finite relative unit quotient over a real quadratic field
{parent=Dirichlet unit theorem}

If $K$ is real quadratic and $L/K$ is a number-field extension with $\mathcal O_L^\times/\mathcal O_K^\times$ finite, then the unit ranks agree. If $d=[L:K]$ and $L$ has signature $(r_1,r_2)$, this forces
$$
r_1+r_2=2,\qquad r_1+2r_2=2d.
$$
For a proper extension the only possibility is $d=2$ and signature $(0,2)$. For $K=\mathbb Q(\sqrt5)$, the example $L=K(i)$ has this signature and a nontrivial finite quotient because $i$ is a new root of unity.

= Logarithmic embedding of number field units
{parent=Dirichlet unit theorem}

Choose the real embeddings $\sigma_i$ and one embedding $\tau_j$ from each complex-conjugate pair. The weighted logarithmic embedding is
$$
\operatorname{Log}(u)=
\bigl(\log|\sigma_1u|,\ldots,\log|\sigma_ru|,
2\log|\tau_1u|,\ldots,2\log|\tau_su|\bigr).
$$
For a unit, the sum of these coordinates is $\log|N_{K/\mathbb Q}(u)|=0$.

= Kernel of the logarithmic unit embedding
{parent=Logarithmic embedding of number field units}

The kernel of $\operatorname{Log}:\mathcal O_K^\times\to\mathbb R^{r+s}$ is exactly $\mu(K)$. If every conjugate of a unit has modulus one, all its powers lie in a bounded subset of the discrete Minkowski lattice $\mathcal O_K$, so two powers coincide and the unit is a root of unity.

= Dirichlet unit lattice
{parent=Logarithmic embedding of number field units}
{c}

The logarithms of a fundamental system of $r+s-1$ units form a basis of the hyperplane
$$
H=\{x\in\mathbb R^{r+s}:x_1+\cdots+x_{r+s}=0\},
$$
and their integer span is a full lattice in $H$.

= Archimedean balancing of a principal ideal generator
{parent=Dirichlet unit lattice}

Every principal ideal $I$ of a number field has a generator $\alpha$ satisfying
$$
|\sigma(\alpha)|\leq C_KN(I)^{1/[K:\mathbb Q]}
$$
at every archimedean embedding. Multiply an arbitrary generator by a unit so that its logarithmic unit-lattice coordinates lie in a fixed bounded fundamental parallelepiped.

= Units of Q of square root ten
{parent=Dirichlet unit theorem}

For $K=\mathbb Q(\sqrt{10})$,
$$
\mathcal O_K^\times
=\{\mathord\pm(3+\sqrt{10})^n:n\in\mathbb Z\}.
$$
The fundamental unit $3+\sqrt{10}$ has norm minus one.

= Units of Q of square root minus three
{parent=Dirichlet unit theorem}

For $K=\mathbb Q(\sqrt{-3})$ and $\omega=(-1+\sqrt{-3})/2$,
$$
\mathcal O_K^\times
=\{\mathord\pm1,\mathord\pm\omega,\mathord\pm\omega^2\}
\cong C_6.
$$

= Unit group of the fifth cyclotomic field
{parent=Dirichlet unit theorem}

For a primitive fifth root of unity $\zeta$,
$$
\mathcal O_{\mathbb Q(\zeta)}^\times
\cong C_{10}\times\mathbb Z.
$$
The explicit units $-\zeta$ and $1+\zeta$ generate a subgroup of finite index.

= Field trace
{parent=Algebraic number theory}
{wiki}

The field trace $\operatorname{Tr}_{L/K}(x)$ is the trace of multiplication by $x$, equivalently the sum of its conjugates in a separable extension.

= Trace form of a number field
{title2=$B(x,y)=\operatorname{Tr}_{K/\mathbb Q}(xy)$}
{parent=Field trace}
{wiki=Trace_form}

The trace form of a number field $K$ is the symmetric bilinear form
$$
B(x,y)=\operatorname{Tr}_{K/\mathbb Q}(xy).
$$

= Nondegeneracy of the trace form of a number field
{parent=Trace form of a number field}

For nonzero $x\in K$, choose $y=x^{-1}$. Then
$$
B(x,y)=\operatorname{Tr}_{K/\mathbb Q}(1)=[K:\mathbb Q]\ne0,
$$
so the trace form is <nondegenerate bilinear form>[nondegenerate].

= Trace and norm integrality criterion for a quadratic extension
{parent=Field trace}

In a quadratic extension $L/K$, an element is integral exactly when its relative trace and norm lie in $\mathcal O_K$.

= Fractional ideal
{parent=Algebraic number theory}
{wiki}

A fractional ideal of a number field is a nonzero $\mathcal O_K$-submodule $I\subset K$ for which $dI\subseteq\mathcal O_K$ for some nonzero $d\in\mathcal O_K$.

= Integral ideal
{parent=Fractional ideal}

An integral ideal is a fractional ideal contained in the ring of integers $\mathcal O_K$.

= Ideal norm
{parent=Algebraic number theory}
{wiki}

For a nonzero integral ideal $I\subseteq\mathcal O_K$, its norm is the finite index $N(I)=|\mathcal O_K/I|$. Norms multiply under ideal multiplication.

= Ideal class group
{parent=Algebraic number theory}
{wiki}

The ideal class group is the group of nonzero fractional ideals modulo principal fractional ideals.

= Minkowski bound for ideal classes
{parent=Ideal class group}
{c}
{wiki=Minkowski%27s_bound}

If a number field of degree $n$ and signature $(r_1,r_2)$ has discriminant $d_K$, every ideal class contains an integral ideal $I$ with
$$
N(I)\leq
\left(\frac4\pi\right)^{r_2}
\frac{n!}{n^n}\sqrt{|d_K|}.
$$

= Class group of Q of cube root of three
{title2=$\operatorname{Cl}(\mathbb Q(\sqrt[3]{3}))$}
{parent=Minkowski bound for ideal classes}
{c}

For $K=\mathbb Q(\sqrt[3]{3})$ with $\mathcal O_K=\mathbb Z[\sqrt[3]{3}]$, the field discriminant is $-243$ and the Minkowski bound is $8\sqrt3/\pi<5$. The prime ideals above $2$ and $3$ are principal, so every ideal class has a principal representative and
$$
\operatorname{Cl}(K)=1.
$$

= Ideal class group of Q of square root of minus thirty-three
{title2=$\operatorname{Cl}(\mathbb Q(\sqrt{-33}))$}
{parent=Minkowski bound for ideal classes}
{c}

The field has ring of integers $\mathbb Z[\sqrt{-33}]$, discriminant $-132$, and Minkowski bound below eight. The nonprincipal ramified ideals above two and three generate four classes, and the split primes above seven introduce no new class. Hence
$$
\operatorname{Cl}(\mathbb Q(\sqrt{-33}))
\cong C_2\times C_2.
$$

= Order of an ideal class
{parent=Ideal class group}

The order of the class of a fractional ideal $I$ is the least positive integer $n$ for which $I^n$ is principal.

= Odd-order obstruction for a split prime in an imaginary quadratic field
{parent=Order of an ideal class}

Let $p\equiv11\pmod{12}$, $K=\mathbb Q(\sqrt{-p})$, and let $\mathfrak p$ be a prime-ideal factor of $(3)$. If $p>3^{n+2}$ and $n$ is odd, then $[\mathfrak p]$ does not have order $n$. Otherwise $\mathfrak p^n=(\alpha)$ would give
$$
4\cdot3^n=a^2+pb^2
$$
for $\alpha=(a+b\sqrt{-p})/2$. The bound forces $b=0$, after which $3^n=N(\alpha)$ is an integer square, impossible for odd $n$.

= Bounded-norm ideal representatives
{parent=Ideal class group}

If every integral ideal $I$ contains nonzero $\alpha$ with $|N(\alpha)|\leq C N(I)$, then $(\alpha)I^{-1}$ is an integral representative of $[I]^{-1}$ with norm at most $C$.

= Finiteness of the ideal class group
{parent=Ideal class group}
{wiki=Ideal_class_group\#Finiteness_of_the_class_group}

Bounded-norm representatives prove class-group finiteness because only finitely many integral ideals have norm at most a fixed constant.

= Ideal classes are torsion
{parent=Ideal class group}

Every element of a finite ideal class group has finite order. Therefore every nonzero ideal has a positive power that is principal.

= Unique factorization of ideals in a number field
{parent=Algebraic number theory}
{wiki=Dedekind_domain\#Unique_factorization_of_ideals}

Every nonzero proper ideal of a number ring factors uniquely as a finite product of positive powers of distinct nonzero prime ideals.

= Dedekind factorization theorem
{parent=Unique factorization of ideals in a number field}
{c}
{wiki=Dedekind%27s_theorem}

If $\mathcal O_K=\mathbb Z[\theta]$ and the minimal polynomial of $\theta$ factors modulo a rational prime $p$ as
$$
\overline f=\prod_i\overline g_i^{e_i},
$$
then
$$
(p)=\prod_i(p,g_i(\theta))^{e_i},
$$
and the prime ideal $(p,g_i(\theta))$ has norm $p^{\deg g_i}$.

= Dedekind zeta function
{parent=Algebraic number theory}
{c}
{wiki}

For a number field $K$, the Dedekind zeta function is formally
$$
\zeta_K(s)=\sum_{0\ne\mathfrak a\subseteq\mathcal O_K}N(\mathfrak a)^{-s}.
$$

= Euler product for the Dedekind zeta function
{parent=Dedekind zeta function}

Unique factorization of nonzero ideals and multiplicativity of the ideal norm give the formal identity
$$
\zeta_K(s)=\prod_{\mathfrak p}
\left(1-N(\mathfrak p)^{-s}\right)^{-1},
$$
where the product ranges over nonzero prime ideals.

= Quadratic Dedekind zeta factorization
{parent=Dedekind zeta function}

For a quadratic field $K$ of discriminant $D$,
$$
\zeta_K(s)=\zeta_{\mathbb Q}(s)L(\chi_D,s),
$$
where $\chi_D(p)$ is $1$, $-1$, or $0$ according as $(p)$ splits, remains prime, or ramifies in $\mathcal O_K$. Comparing the three possible local Euler factors proves the formal identity.

= Primary ideal
{parent=Algebraic number theory}
{wiki}

An ideal $I$ is primary when $ab\in I$ and $a\notin I$ imply $b^n\in I$ for some positive integer $n$.

= Radical of a primary ideal
{parent=Primary ideal}

The radical $\sqrt I=\{x:x^n\in I\text{ for some }n\geq1\}$ of a primary ideal is prime. In a number ring this forces a nonzero primary ideal to have only one prime in its unique factorization.

= Galois theory
{c}
{parent=Algebra}
{wiki}

= Frobenius endomorphism
{title2=$x\mapsto x^p$}
{parent=Galois theory}
{c}
{wiki}

In a field of characteristic $p>0$, the Frobenius map $x\mapsto x^p$ is an injective field endomorphism because
$$
(x+y)^p=x^p+y^p.
$$
Consequently every element of an extension field has at most one $p$th root.

= Formal derivative in positive characteristic
{title2=$f'(T)$}
{parent=Galois theory}
{wiki=Formal_derivative}

The formal derivative of $f(T)=\sum_i a_iT^i$ is $f'(T)=\sum_i ia_iT^{i-1}$. In characteristic $p$, the coefficient $i$ vanishes exactly when $p$ divides $i$.

= Polynomial with zero formal derivative in characteristic p
{parent=Formal derivative in positive characteristic}

Over a field of characteristic $p>0$,
$$
f'(T)=0
\quad\Longleftrightarrow\quad
f(T)=g(T^p)
$$
for some polynomial $g$. Indeed, zero derivative says every exponent with nonzero coefficient is divisible by $p$.

= Separable polynomial
{parent=Galois theory}
{wiki=Separable_polynomial}

A polynomial over a field is separable when it has no repeated root in a splitting field. Equivalently, it is coprime to its <formal derivative in positive characteristic>[formal derivative].

= Automorphism-count divisibility theorem
{parent=Galois theory}

For a finite extension $K/F$, the automorphism group $G=\operatorname{Aut}_F(K)$ has order dividing $[K:F]$. Indeed, Artin's theorem gives $[K:K^G]=|G|$, while $F\subseteq K^G$ and the tower law gives the divisibility.

= Coprime-degree descent for powers
{parent=Galois theory}

Let $L/K$ have degree $d$ with $\gcd(d,n)=1$. If $a=x^n\in K$ for some $x\in L$, then
$$
a^d=N_{L/K}(x)^n.
$$
Choosing $r,s\in\mathbb Z$ with $rd+sn=1$ gives
$$
a=\bigl(N_{L/K}(x)^r a^s\bigr)^n,
$$
so $a$ was already an $n$th power in $K$.

= Algebraic element
{parent=Galois theory}
{wiki}

An element $\alpha$ of a field extension $L/K$ is algebraic over $K$ when some nonzero polynomial in $K[X]$ vanishes at $\alpha$.

= Minimal polynomial of an algebraic element
{title2=$m_{\alpha,K}(X)$}
{parent=Algebraic element}
{wiki=Minimal_polynomial_(field_theory)}

The minimal polynomial of an algebraic element $\alpha$ over $K$ is the unique monic polynomial of least positive degree in $K[X]$ that vanishes at $\alpha$. It is irreducible, and its degree equals $[K(\alpha):K]$.

= Primitive element of a field extension
{parent=Galois theory}
{wiki=Primitive_element}

A primitive element of a finite extension $L/K$ is an element $\alpha\in L$ such that $L=K(\alpha)$.

= Primitive element theorem
{parent=Galois theory}
{wiki}

Every finite separable extension $L/K$ is generated by one element: $L=K(\alpha)$ for some $\alpha\in L$.

= Finite separable simple extension implications
{parent=Galois theory}

A finite separable extension is simple by the primitive element theorem. A simple separable extension is finite because its generator is separable and hence algebraic. Finite and simple does not imply separable: in characteristic $p$, $\mathbb F_p(t^{1/p})/\mathbb F_p(t)$ is finite and simple but purely inseparable.

= Biquadratic extension
{parent=Galois theory}
{wiki}

If $a,b,ab$ are nonsquares in a field $K$ of characteristic other than two, then
$$
K(\sqrt a,\sqrt b)/K
$$
is a degree-four Galois extension with group $C_2\times C_2$. Its three quadratic intermediate fields are $K(\sqrt a)$, $K(\sqrt b)$, and $K(\sqrt{ab})$.

= Transitive subgroups of the symmetric group on four points
{parent=Galois theory}

Up to conjugacy, the transitive subgroups of $S_4$ are $C_4$, $V_4$, $D_8$, $A_4$, and $S_4$.

= Dihedral Galois action on four radical roots
{parent=Galois theory}

If $r$ cyclically sends $\alpha\mapsto\beta\mapsto-\alpha\mapsto-\beta$ and $s$ fixes $\alpha$ while negating $\beta$, then $r^4=s^2=1$ and $srs=r^{-1}$. These automorphisms realize $D_8$ as the Galois group of the corresponding quartic splitting field.

= Field embedding
{parent=Galois theory}
{wiki=Field_homomorphism}

A field embedding $L\to\Omega$ is an injective field homomorphism. A $K$-embedding fixes every element of the base field $K$.

= Linear independence of distinct field embeddings
{parent=Field embedding}
{wiki=Dedekind%27s_independence_of_characters}

Distinct field embeddings $\sigma_1,\ldots,\sigma_n:L\to\Omega$ are linearly independent over $\Omega$ as functions on $L$. Thus a nontrivial linear combination $\sum_i a_i\sigma_i$ cannot vanish on every element of $L$.

= Lagrange resolvent eigenvector for a cyclic field automorphism
{parent=Linear independence of distinct field embeddings}

If $\sigma$ has order $n$ and the fixed field contains a primitive $n$th root $\zeta$, then
$$
T_\zeta(\beta)=\sum_{j=0}^{n-1}\zeta^{-j}\sigma^j(\beta)
$$
satisfies $\sigma(T_\zeta(\beta))=\zeta T_\zeta(\beta)$. Independence of the powers of $\sigma$ ensures that $T_\zeta(\beta)$ is nonzero for some $\beta$.

= Extension count for field embeddings
{parent=Field embedding}

An embedding of $K(\alpha_1,\ldots,\alpha_i)$ extends across $\alpha_{i+1}$ by sending it to a root of its transformed minimal polynomial. There are between one and the relative degree many choices, with equality for a separable extension.

= Tower law for field extensions
{parent=Galois theory}
{wiki=Degree_of_a_field_extension\#The_multiplicativity_formula_for_degrees}

For finite extensions $K\subseteq E\subseteq L$, the degrees multiply: $[L:K]=[L:E][E:K]$.

= Splitting field
{parent=Galois theory}
{wiki}

The splitting field of a polynomial over $K$ is the smallest field extension of $K$ in which the polynomial is a product of linear factors.

= Existence and uniqueness of splitting fields
{parent=Splitting field}

Every nonconstant polynomial $f\in K[X]$ has a splitting field of finite degree over $K$. It is constructed by adjoining a root of an irreducible factor and repeating over the enlarged field. Any two splitting fields of $f$ over $K$ are isomorphic by an isomorphism that fixes $K$. Inside one fixed algebraic closure, the field generated by all roots is unique as a subfield.

= Splitting field of x cubed minus two
{title2=$\mathbb Q(\sqrt[3]{2},\zeta_3)$}
{parent=Splitting field}

If $\alpha=\sqrt[3]{2}$ is the positive real root and $\zeta_3=e^{2\pi i/3}$, then the roots of $X^3-2$ are
$$
\alpha,\qquad \zeta_3\alpha,\qquad \zeta_3^2\alpha.
$$
Their splitting field over $\mathbb Q$ is $\mathbb Q(\alpha,\zeta_3)$.

= Finite normal extension as a splitting field
{parent=Splitting field}

A finite extension $L/K$ is <normal field extension>[normal] exactly when it is the splitting field over $K$ of some polynomial. For the forward direction, multiply the minimal polynomials of finitely many generators of $L$; normality puts all their roots in $L$. The reverse direction follows because a $K$-embedding permutes the roots of a polynomial and hence preserves its splitting field.

= Splitting field of a polynomial obtained by Frobenius substitution
{parent=Splitting field}

Let $g\in K[T]$ in characteristic $p$, let $f(T)=g(T^p)$, let $L$ split $g$ over $K$, and let $M$ split $f$ over $L$. If $y_i$ are the roots of $g$ and $x_i^p=y_i$, then the <Frobenius endomorphism> makes each $x_i$ unique and
$$
M=L(x_i:i)=K(x_i:i),
$$
because $y_i=x_i^p$. Thus $M$ is already the splitting field of $f$ over $K$, and every root of $f$ is purely inseparable over $L$.

= Normal field extension
{parent=Galois theory}
{wiki=Normal_extension}

An algebraic extension $L/K$ is normal when every irreducible polynomial over $K$ with a root in $L$ splits in $L$. For finite extensions, equivalently every $K$-embedding into an algebraic closure has image $L$.

= Separable field extension
{parent=Galois theory}
{wiki=Separable_extension}

An algebraic extension is separable when every element has a minimal polynomial with distinct roots.

= Separable algebraic element
{parent=Separable field extension}

An algebraic element $x$ over $K$ is separable when its <minimal polynomial> over $K$ has no repeated root in a splitting field. For an irreducible polynomial $m_x$, this is equivalent to $m_x'\ne0$, because then $\gcd(m_x,m_x')=1$.

= Purely inseparable field extension
{parent=Galois theory}
{wiki=Purely_inseparable_extension}

An algebraic extension in characteristic $p>0$ is purely inseparable when every one of its elements is purely inseparable over the base field.

= Purely inseparable algebraic element
{parent=Purely inseparable field extension}

An algebraic element $x$ is purely inseparable over $K$ when $x^{p^n}\in K$ for some $n\geq0$.

= Minimal polynomial of a purely inseparable element
{parent=Purely inseparable algebraic element}

An element is purely inseparable over $K$ exactly when its minimal polynomial has the form
$$
T^{p^n}-a,
\qquad a\in K.
$$
To prove the nontrivial direction, repeatedly factor a zero formal derivative through $T\mapsto T^p$ until the remaining irreducible polynomial has nonzero derivative. Since $T^{p^r}-x^{p^r}$ has only the root $x$, that remaining separable polynomial must be linear.

= Unique embedding extension through a purely inseparable extension
{parent=Purely inseparable field extension}

If $M/L$ is purely inseparable, an embedding of $L$ into an algebraic closure has at most one extension to $M$: if $x^{p^n}\in L$, the image of $x$ must be the unique $p^n$th root of the image of $x^{p^n}$. Existence in a normal overfield therefore implies uniqueness.

= Finite Galois extension
{parent=Galois theory}
{c}
{wiki=Galois_extension}

A finite extension $L/K$ is Galois when it has $[L:K]$ automorphisms fixing $K$.

= Normal and separable implies Galois
{parent=Finite Galois extension}

A separable extension has $[L:K]$ embeddings in an algebraic closure, and normality makes every one an automorphism of $L$. Hence a finite normal separable extension is Galois.

= Roots of unity in a finite field
{parent=Galois theory}
{wiki=Root_of_unity_modulo_n}

The multiplicative group $\mathbb F_q^\times$ is cyclic of order $q-1$, so it contains all $m$th roots of unity when $m$ divides $q-1$ and the characteristic does not divide $m$.

= Kummer extension of rational function fields
{parent=Galois theory}
{c}

If a field $k$ has characteristic prime to $m$ and contains all $m$th roots of unity, then $k(X)/k(X^m)$ is a cyclic Galois extension of degree $m$, with automorphisms $X\mapsto\zeta X$.

= Galois group of a polynomial
{c}
{parent=Galois theory}
{wiki=Galois_group}

The Galois group of a separable polynomial acts faithfully on its roots; this action is transitive exactly when the polynomial is irreducible.

= Transitivity lifted through unique pth roots
{parent=Galois group of a polynomial}

Suppose a group of field automorphisms acts transitively on elements $y_i$, and each $y_i$ has a unique $p$th root $x_i$ in a stable extension. If every automorphism extends, it sends $x_i$ to the unique root above its image of $y_i$, so the extended group acts transitively on the $x_i$.

= Irreducible factors after Frobenius substitution
{parent=Galois theory}

Let $g\in K[T]$ be irreducible and separable in characteristic $p$, and put $f(T)=g(T^p)$. Every monic irreducible factor $h$ of $f$ is either $f$ itself or separable. Indeed, if $h$ were inseparable, then $h(T)=q(T^p)$; divisibility of $q(T^p)$ into $g(T^p)$ forces $q\mid g$, hence $q=g$ and $h=f$.

= Reducibility criterion after Frobenius substitution
{parent=Irreducible factors after Frobenius substitution}

Under the same hypotheses, $g(T^p)$ is reducible exactly when every coefficient of $g$ is a $p$th power in $K$. If $g=\sum a_i^pT^i$, then
$$
g(T^p)=\left(\sum a_iT^i\right)^p.
$$
Conversely, if $g(T^p)$ is reducible, all its irreducible factors are separable. Its zero derivative forces every factor multiplicity to be divisible by $p$, so it is a $p$th power in $K[T]$ and all its coefficients are $p$th powers.

= Frobenius cycle type
{c}
{parent=Galois theory}

Over $\mathbb F_q$, Frobenius acts on roots of an irreducible degree-$n$ polynomial as an $n$-cycle. Factor degrees therefore give its cycle type.

= Dedekind factorization cycle test
{c}
{parent=Galois theory}

At an unramified prime, the degrees of the irreducible factors of a polynomial modulo that prime give the cycle type of an element of the Galois group.

= Cyclotomic polynomial
{parent=Galois theory}
{wiki}

$\Phi_n$ is the monic polynomial whose roots are the primitive $n$th roots of unity. It belongs to $\mathbb Z[X]$ and is irreducible over $\mathbb Q$.

= Cyclotomic factorization
{parent=Cyclotomic polynomial}

Partitioning the roots of $X^n-1$ by their exact orders gives
$$
X^n-1=\prod_{d\mid n}\Phi_d(X).
$$
Induction and Gauss's lemma then show that every $\Phi_n$ has integer coefficients.

= Separability of a cyclotomic polynomial modulo p
{parent=Cyclotomic polynomial}
{c}

If the prime $p$ does not divide $n$, then $X^n-1$ is square-free over $\mathbb F_p$ because its derivative $nX^{n-1}$ is coprime to it. Its factor $\Phi_n$ is therefore separable modulo $p$.

= Galois embedding for a cyclotomic polynomial
{parent=Cyclotomic polynomial}

If $L$ is the splitting field of $\Phi_n$ over a characteristic-zero field $K$, choosing a primitive root $\zeta_n$ gives an injection
$$
\operatorname{Gal}(L/K)\hookrightarrow(\mathbb Z/n\mathbb Z)^\times,
\qquad
\sigma(\zeta_n)=\zeta_n^{a_\sigma}.
$$
It is surjective exactly when $\Phi_n$ is irreducible over $K$.

= Cyclotomic field
{parent=Cyclotomic polynomial}
{wiki}

A cyclotomic field is generated by a root of unity. If $K\subseteq\mathbb C$ and $\zeta_n$ is primitive, every $K$-conjugate of $\zeta_n$ is a power $\zeta_n^a$ and already lies in $K(\zeta_n)$. Thus $K(\zeta_n)/K$ is Galois, and its Galois group embeds in $(\mathbb Z/n\mathbb Z)^\times$, so it is abelian.

= Roots of unity in a rational cyclotomic field
{parent=Cyclotomic field}

The roots of unity in $\mathbb Q(\zeta_n)$ form a cyclic group of order
$$
\operatorname{lcm}(2,n)=
\begin{cases}
n,&n\text{ even},\\
2n,&n\text{ odd}.
\end{cases}
$$

= Seventh cyclotomic field
{title2=$\mathbb Q(\zeta_7)$}
{parent=Cyclotomic field}

For a primitive seventh root of unity $\zeta_7$,
$$
\operatorname{Gal}(\mathbb Q(\zeta_7)/\mathbb Q)
\cong(\mathbb Z/7\mathbb Z)^\times\cong C_6,
$$
where $\sigma_a(\zeta_7)=\zeta_7^a$. Its four intermediate fields correspond to the four subgroups of the cyclic group $C_6$.

= Quadratic Gaussian period in the seventh cyclotomic field
{title2=$\eta=\zeta_7+\zeta_7^2+\zeta_7^4$}
{parent=Seventh cyclotomic field}
{c}

The unique quadratic subfield of $\mathbb Q(\zeta_7)$ is
$$
\mathbb Q(\eta)=\mathbb Q(\sqrt{-7}),
\qquad
\eta=\frac{-1+\sqrt{-7}}2.
$$
The conjugate period is $-1-\eta$, and
$$
m_{\eta,\mathbb Q}(X)=X^2+X+2.
$$

= Real cubic subfield of the seventh cyclotomic field
{title2=$\theta=\zeta_7+\zeta_7^{-1}$}
{parent=Seventh cyclotomic field}

The unique cubic subfield is the maximal real subfield
$$
\mathbb Q(\theta),
\qquad
\theta=\zeta_7+\zeta_7^{-1},
$$
and
$$
m_{\theta,\mathbb Q}(X)=X^3+X^2-2X-1.
$$
It is the fixed field of complex conjugation.

= Maximal cyclotomic extension
{parent=Cyclotomic field}

For a field $K$ inside a fixed algebraic closure, the maximal cyclotomic extension is the union of the fields $K(\zeta_n)$ generated by roots of unity.

= Maximal cyclotomic extension of the real numbers
{parent=Maximal cyclotomic extension}

The maximal cyclotomic extension of $\mathbb R$ is $\mathbb C$, since adjoining a primitive fourth root of unity adjoins $i$.

= Maximal cyclotomic extension of the rational numbers
{parent=Maximal cyclotomic extension}

The maximal cyclotomic extension of $\mathbb Q$ is a proper subfield of $\overline{\mathbb Q}$. Every finite subextension inside a cyclotomic field is Galois and abelian, whereas $\mathbb Q(\sqrt[3]{2})/\mathbb Q$ is not normal.

= Maximal cyclotomic extension of a finite field
{parent=Maximal cyclotomic extension}

The maximal cyclotomic extension of $\mathbb F_p$ is $\overline{\mathbb F}_p$. A primitive $(p^m-1)$th root of unity generates $\mathbb F_{p^m}$ over $\mathbb F_p$, and these finite fields exhaust the algebraic closure.

= Constructible field extension
{parent=Galois theory}
{wiki=Constructible_number}

A finite real extension $K/\mathbb Q$ is constructible when it is contained in a tower
$$
\mathbb Q=K_0\subset K_1\subset\cdots\subset K_r,
\qquad [K_i:K_{i-1}]=2.
$$
Equivalently, every element of $K$ can be obtained from rational numbers by field operations and successive square roots.

= Constructibility of the real seventeenth cyclotomic field
{parent=Constructible field extension}

The real cyclotomic field
$$
\mathbb Q(\zeta_{17}+\zeta_{17}^{-1})
$$
is Galois of degree eight with cyclic Galois group $(\mathbb Z/17\mathbb Z)^\times/\{\mathord\pm1\}$. A subgroup chain of successive index two gives a tower of three quadratic extensions, proving that $\cos(2\pi/17)$ is constructible.

= Fixed field
{parent=Galois theory}
{wiki}

For a group $G$ of automorphisms of a field $L$, the fixed field is
$$
L^G=\{x\in L:g(x)=x\text{ for every }g\in G\}.
$$
For one automorphism $\sigma$, it is also written $L^\sigma$.

= Artin fixed-field theorem
{c}
{parent=Galois theory}
{wiki=Artin%27s_theorem}

If a finite group $G$ acts faithfully by automorphisms on a field $L$, then $L/L^G$ is Galois, has degree $|G|$, and has Galois group $G$.

= Galois correspondence
{c}
{parent=Galois theory}
{wiki}

For a finite Galois extension, subgroups correspond in reverse order to intermediate fields by taking fixed fields and field-fixing subgroups.

= Normal subextension criterion
{parent=Galois correspondence}

For a finite Galois extension $L/K$ with group $G$, an intermediate extension $E/K$ is normal exactly when its corresponding subgroup $\operatorname{Gal}(L/E)$ is a <normal subgroup> of $G$.

= Subextensions of an abelian Galois extension
{parent=Galois correspondence}

Every subgroup of an abelian group is normal. Hence every intermediate field of a finite abelian Galois extension is itself Galois over the base field.

= Cyclic Galois extension of the rational numbers of every finite degree
{parent=Subextensions of an abelian Galois extension}

For $r>1$, choose a prime $\ell\equiv1\pmod r$ by Dirichlet's theorem. The cyclic group
$$
\operatorname{Gal}(\mathbb Q(\zeta_\ell)/\mathbb Q)
\cong C_{\ell-1}
$$
has a subgroup of index $r$, whose fixed field is cyclic Galois of degree $r$ over $\mathbb Q$.

= Finite extensions of the fixed field of a finite-order automorphism of an algebraically closed field
{parent=Subextensions of an abelian Galois extension}

Let an automorphism $\sigma$ of an algebraically closed field $L$ have finite order and fixed field $K$. Artin's fixed-field theorem makes $L/K$ finite cyclic Galois. Every finite extension $M/K$ embeds in $L$: in characteristic zero, choose a primitive generator and send it to a root in $L$. Its image is an intermediate field of a cyclic extension and is therefore Galois and cyclic over $K$.

= Subfields of the splitting field of x to the fourth minus seven
{parent=Galois correspondence}

Put $\alpha=\sqrt[4]{7}$ and $L=\mathbb Q(\alpha,i)$. The <dihedral group> $D_8=\langle r,s:r^4=s^2=1,\ srs=r^{-1}\rangle$ acts by
$$
r(\alpha)=i\alpha,quad r(i)=i,qquad
s(\alpha)=\alpha,quad s(i)=-i.
$$
The ten subgroups and their fixed fields are
$$
\begin{array}{c|c}
D_8&\mathbb Q\\
\langle r\rangle&\mathbb Q(i)\\
\langle r^2,s\rangle&\mathbb Q(\sqrt7)\\
\langle r^2,rs\rangle&\mathbb Q(\sqrt{-7})\\
\langle r^2\rangle&\mathbb Q(\sqrt7,i)\\
\langle s\rangle&\mathbb Q(\alpha)\\
\langle r^2s\rangle&\mathbb Q(i\alpha)\\
\langle rs\rangle&\mathbb Q((1+i)\alpha)\\
\langle r^3s\rangle&\mathbb Q((1-i)\alpha)\\
\{1\}&L.
\end{array}
$$
The normal subgroups are $D_8$, $\langle r\rangle$, the two displayed Klein four-groups, $\langle r^2\rangle$, and $\{1\}$.

= Polynomial discriminant
{parent=Galois theory}
{wiki=Discriminant\#Polynomial_discriminant}

For a monic polynomial with roots $r_i$, the discriminant is $\prod_{i<j}(r_i-r_j)^2$ and vanishes exactly for a repeated root.

= Vandermonde determinant
{parent=Polynomial discriminant}
{c}
{wiki}

The Vandermonde product $\prod_{i<j}(r_i-r_j)$ is alternating, and its square is the discriminant of the monic polynomial with roots $r_i$.

= Discriminant of a depressed cubic
{parent=Polynomial discriminant}

The depressed cubic $x^3+ax+b$ has discriminant $-4a^3-27b^2$.

= Discriminant criterion for an alternating Galois group
{parent=Polynomial discriminant}

For a monic separable polynomial over a field of characteristic other than two, its Galois group is contained in the alternating group exactly when its discriminant is a square in the base field. The Galois action multiplies the Vandermonde product by the sign of the induced root permutation.

= Lagrange resolvent for a cubic
{parent=Galois theory}
{c}
{wiki=Lagrange_resolvent}

For cubic roots $\alpha_i$ and a cube root of unity $\omega$, the Fourier combinations $u=\alpha_1+\omega\alpha_2+\omega^2\alpha_3$ and $v$ have cubes satisfying a quadratic resolvent.

= Cubic resolvent
{parent=Lagrange resolvent for a cubic}

For $x^3+ax+b$, the Lagrange-resolvent cubes are roots of $X^2+27bX-27a^3$.

= Cardano formula
{parent=Cubic resolvent}
{c}
{wiki}

Cardano's formula solves a depressed cubic by taking square roots in its quadratic resolvent and then compatible cube roots.

= Solvability by radicals
{parent=Galois theory}
{wiki}

A polynomial is solvable by radicals when its roots lie in a tower obtained by adjoining successive roots; equivalently, its Galois group is solvable.

= Galois group of an irreducible cubic
{parent=Galois theory}
{c}

An irreducible cubic over a field of characteristic other than two has Galois group $A_3$ when its discriminant is a square and $S_3$ otherwise.

= Splitting field of x cubed minus x minus one
{parent=Galois group of an irreducible cubic}

The polynomial $X^3-X-1$ is irreducible over $\mathbb Q$ and has discriminant $-23$, so its splitting field has Galois group $S_3$. Its unique quadratic subfield is the fixed field of $A_3$, namely $\mathbb Q(\sqrt{-23})$.

= Cubic discriminant criterion
{parent=Galois group of an irreducible cubic}

The discriminant square class detects whether the transitive cubic Galois group lies in the alternating group.

= Characteristic-two cubic resolvent
{parent=Galois theory}

Let $x_1,x_2,x_3$ be the roots of the separable polynomial $T^3+aT+b$ in characteristic two. The cyclic sums
$$
r=x_1^2x_2+x_2^2x_3+x_3^2x_1,
\qquad
s=x_2^2x_1+x_3^2x_2+x_1^2x_3
$$
are the distinct roots of
$$
T^2+bT+a^3+b^2.
$$
Even root permutations fix $r,s$, while odd permutations interchange them. The cubic Galois group is therefore contained in $A_3$ exactly when this quadratic splits over the base field.

= Representation theory
{parent=Algebra}
{wiki}

= Group representation
{parent=Representation theory}
{wiki=Representation_(mathematics)}

A group representation is a <group homomorphism> from a <group> to the <general linear group> of a <vector space>. It realizes abstract group elements as invertible linear transformations.

= Semisimple representation
{parent=Group representation}
{wiki=Semisimple_representation}

A representation is semisimple when it is a <direct sum> of <irreducible representation>[irreducible representations]. Equivalently, every invariant subspace has an invariant complement.

= Maschke's theorem
{c}
{parent=Representation theory}
{wiki=Maschke%27s_theorem}

Every <group representation> of a <finite group> over a <field> whose <characteristic of a field>[characteristic] does not divide the number of elements of the group is <semisimple representation>[semisimple]. Given an invariant subspace, averaging any projection onto it over the group produces an equivariant projection, whose kernel is an invariant complement.

= Complex representation
{parent=Representation theory}
{wiki=Representation_(mathematics)}

A complex representation of a group $G$ is a homomorphism $\rho:G\to\operatorname{GL}(V)$ for a complex vector space $V$.

= Continuous representation of a topological group
{parent=Complex representation}
{wiki=Group_representation#Topological_groups}

A finite-dimensional continuous representation of a topological group is a continuous homomorphism $\rho:G\to\operatorname{GL}(V)$, where $V$ is a finite-dimensional complex vector space.

= Degree of a representation
{parent=Complex representation}

The degree of a finite-dimensional representation $\rho:G\to\operatorname{GL}(V)$ is $\dim V$.

= Isomorphic representations
{parent=Complex representation}

Representations $\rho:G\to\operatorname{GL}(V)$ and $\rho':G\to\operatorname{GL}(V')$ are isomorphic when an invertible linear map $T:V\to V'$ intertwines the actions: $T\rho(g)=\rho'(g)T$ for every $g\in G$.

= Faithful representation
{parent=Complex representation}
{wiki=Faithful_representation}

A representation is faithful when its kernel is trivial.

= Regular representation
{parent=Faithful representation}
{wiki=Regular_representation}

The left regular representation acts on the basis $(e_h)_{h\in G}$ by $\rho(g)e_h=e_{gh}$. It is faithful.

= Faithful irreducible representation of a finite simple group
{parent=Regular representation}

By <Maschke's theorem>, the complex regular representation of a finite group is a direct sum of irreducibles. Its kernel is the intersection of their kernels. If the group is simple, each kernel is either trivial or the whole group; faithfulness of the regular representation therefore forces at least one irreducible constituent to be faithful.

= Spectrum orbit bound for a faithful symmetric-group representation
{parent=Faithful representation}

For a $p$-cycle $g\in S_p$, each $g^k$ with $1\leq k<p$ is conjugate to $g$. In a faithful complex representation, $\rho(g)$ has a nontrivial $p$th-root eigenvalue $\lambda$, and similarity of $\rho(g)$ and $\rho(g)^k$ puts all $p-1$ values $\lambda^k$ in its spectrum. Hence the representation has dimension at least $p-1$.

= Finite-dimensional representation obstruction from elementary abelian subgroups
{parent=Faithful representation}

Commuting complex involutions are simultaneously diagonalizable, so $(C_2)^n$ can act faithfully in dimension $d$ only if $n\leq d$. A group containing $(C_2)^n$ for arbitrarily large $n$, such as the full permutation group of $\mathbb N$, has no faithful finite-dimensional complex representation.

= Real Heisenberg quotient has no faithful finite-dimensional representation
{parent=Faithful representation}

Let $G$ be the real upper unitriangular three-by-three group, let $Z$ be its centre, and let $Z_0$ be the integer subgroup of $Z$. Every finite-dimensional continuous complex representation of $G/Z_0$ kills the central circle $Z/Z_0$, and hence is not faithful.

= One-dimensional representation kills the commutator subgroup
{parent=Complex representation}

The image of a one-dimensional representation lies in the abelian group $\mathbb C^\times$, so its kernel contains the <commutator subgroup> of the represented group.

= Representation of SU2
{parent=Representation theory}
{wiki=Representation_theory_of_SU(2)}

Finite-dimensional complex representations of $SU_2$ are completely reducible, and their irreducibles are indexed by nonnegative integers.

= Homogeneous polynomial representation of SU2
{parent=Representation of SU2}

The irreducible representation $V_n=\operatorname{Sym}^n(\mathbb C^2)$ consists of homogeneous degree-$n$ polynomials in two variables, with the action induced from the standard two-dimensional representation. It has dimension $n+1$.

= Character of the homogeneous polynomial representation of SU2
{parent=Homogeneous polynomial representation of SU2}

If $g\in SU_2$ has eigenvalues $z,z^{-1}$, then
$$
\chi_{V_n}(g)=z^n+z^{n-2}+\cdots+z^{-n}.
$$

= Classification of finite-dimensional representations of SU2
{parent=Representation of SU2}

Every finite-dimensional complex $SU_2$-representation is a direct sum of the pairwise nonisomorphic irreducibles $V_n=\operatorname{Sym}^n(\mathbb C^2)$.

= Self-duality of finite-dimensional SU2 representations
{parent=Representation of SU2}

The standard invariant alternating form identifies $V_1$ with its dual. Its symmetric powers identify every $V_n$ with $V_n^*$, and complete reducibility then gives $V\cong V^*$ for every finite-dimensional complex $SU_2$-representation.

= Central parity on SU2 tensor products
{parent=Representation of SU2}

On $V_n$, the central element $-I$ acts as $(-1)^n$. It therefore acts trivially on $V_n\otimes V_n$, although it need not act trivially on the tensor square of a representation containing irreducibles of both parities.

= Clebsch-Gordan decomposition for SU2
{parent=Representation of SU2}
{c}
{wiki=Clebsch–Gordan_coefficients}

For $m,n\geq0$,
$$
V_m\otimes V_n\cong
\bigoplus_{j=0}^{\min(m,n)}V_{m+n-2j}.
$$
The identity follows by multiplying the weight characters and comparing their nested weight strings.

= Flip parity in the SU2 tensor square
{parent=Clebsch-Gordan decomposition for SU2}

On the multiplicity-one summand $V_{2n-2j}\subset V_n\otimes V_n$, interchange of tensor factors acts by $(-1)^j$. A highest-weight vector exhibiting the sign is
$$
\sum_{k=0}^j(-1)^k\binom jk
x^{n-k}y^k\otimes x^{n-j+k}y^{j-k}.
$$

= Exterior square of an SU2 irreducible representation
{parent=Flip parity in the SU2 tensor square}

The odd-parity summands give
$$
\bigwedge^2V_n\cong
\bigoplus_{k=0}^{\lfloor(n-1)/2\rfloor}V_{2n-4k-2}.
$$

= Second and third exterior powers of V4 of SU2
{title2=$\bigwedge^2V_4\cong\bigwedge^3V_4\cong V_6\oplus V_2$}
{parent=Exterior square of an SU2 irreducible representation}

The alternating summands in the tensor square give $\bigwedge^2V_4\cong V_6\oplus V_2$. Since $V_4$ is self-dual and has trivial determinant,
$$
\bigwedge^3V_4\cong(\bigwedge^2V_4)^*\otimes\det V_4
\cong V_6\oplus V_2.
$$

= Character of an exterior square
{parent=Representation theory}

For any finite-dimensional complex representation,
$$
\chi_{\wedge^2V}(g)=
\frac{\chi_V(g)^2-\chi_V(g^2)}2.
$$

= Character of an exterior cube
{parent=Representation theory}

Newton's identities applied to the eigenvalues of $g$ give
$$
\chi_{\wedge^3V}(g)
=\frac{\chi_V(g)^3-3\chi_V(g)\chi_V(g^2)+2\chi_V(g^3)}6.
$$

= Averaging over a finite group
{parent=Representation theory}

Averaging any object over a finite group, $|G|^{-1}\sum_{g\in G}g\cdot x$, produces a group-invariant object whenever the relevant operations are linear.

= Unitary representation
{parent=Representation theory}
{wiki}

A complex representation is unitary when every representing matrix preserves a positive-definite Hermitian inner product.

= Unitarization of a finite-group representation
{parent=Unitary representation}

Average a positive-definite Hermitian form over a finite group and choose an orthonormal basis for the averaged form. In that basis every representing matrix is unitary.

= Unitarization of a compact-group representation
{parent=Unitary representation}

For a compact group with normalized <Haar measure>, averaging any positive-definite Hermitian form,
$$
\langle v,w\rangle_G
=\int_G\langle\rho(g)v,\rho(g)w\rangle\,dg,
$$
produces an invariant positive-definite Hermitian form.

= Representation of the circle group
{parent=Unitary representation}
{wiki=Circle_group#Representations}

Every finite-dimensional continuous complex representation of $S^1$ is unitary and decomposes into one-dimensional weight spaces. Its continuous one-dimensional characters are $z\mapsto z^m$ for $m\in\mathbb Z$.

= Central circle weight-space decomposition
{parent=Representation of the circle group}

If a central subgroup is isomorphic to $S^1$, its weight spaces are invariant under the entire group, because every representing operator commutes with the central circle action.

= Character orthogonality
{parent=Representation theory}

Irreducible characters are orthonormal for $\langle\chi,\psi\rangle=|G|^{-1}\sum_g\chi(g)\overline{\psi(g)}$, and $\sum_\chi\chi(1)^2=|G|$.

= Character table
{parent=Representation theory}
{wiki}

The character table of a finite group lists its irreducible complex characters on its conjugacy classes. The number of rows equals the number of conjugacy classes, and the squared row degrees sum to the group order.

= Irreducible character degree
{parent=Character table}

The degree of an irreducible character $\chi$ is $\chi(1)$, the dimension of the corresponding irreducible representation.

= Irreducible character degrees
{synonym}

= Linear character
{parent=Irreducible character degree}
{wiki=Character_theory}

A linear character is the character of a one-dimensional representation and therefore has degree one.

= Linear characters
{synonym}

= Nonlinear irreducible character
{parent=Irreducible character degree}

A nonlinear irreducible character has degree greater than one.

= Character of a representation
{parent=Representation theory}
{wiki=Character_theory}

The character of a finite-dimensional representation $\rho$ is the class function $\chi_\rho(g)=\operatorname{tr}(\rho(g))$. Isomorphic representations have the same character.

= Burnside lemma
{c}
{parent=Representation theory}
{wiki=Burnside%27s_lemma}

For a finite group $G$ acting on a finite set $X$, the number of orbits is
$$
|X/G|=\frac1{|G|}\sum_{g\in G}|X^g|.
$$
It follows by counting pairs $(g,x)$ with $gx=x$ first by $g$ and then by $x$.

= Permutation representation of a two-transitive action
{parent=Burnside lemma}

If a finite group acts two-transitively on $X$, then
$$
\mathbb C[X]\cong\mathbf1\oplus V,
$$
where $V$ is irreducible and nontrivial. The character norm is the number of orbits on $X\times X$, namely the diagonal and its complement.

= Permutation representation
{parent=Representation theory}
{wiki}

For a finite $G$-set $X$, the character of $\mathbb C[X]$ at $g$ is $|X^g|$. Its invariant dimension is the number of $G$-orbits.

= Symmetric-group subset permutation representation
{parent=Permutation representation}

Let $S_n$ act on the $r$-element subsets of $\{1,\ldots,n\}$, with permutation character $\pi_r$. If $0\leq l\leq k\leq n/2$, then
$$
\langle\pi_k,\pi_l\rangle=l+1,
$$
because orbits of pairs are classified by intersection size $0,\ldots,l$.

= Inclusion map between subset permutation modules
{parent=Symmetric-group subset permutation representation}

For $r\leq n/2$, the equivariant map from $(r-1)$-subsets to $r$-subsets that sends each subset to the sum of the $r$-subsets containing it is injective over $\mathbb C$. Hence $\pi_r-\pi_{r-1}$ is the character of a representation.

= Standard representation of the symmetric group
{parent=Permutation representation}
{wiki=Standard_representation}

The permutation representation of $S_n$ on $\mathbb C^n$ decomposes as
$$
\mathbb C(1,\ldots,1)
\oplus
\{(x_1,\ldots,x_n):x_1+\cdots+x_n=0\}.
$$
The first summand is the trivial representation and the second is the $(n-1)$-dimensional standard representation.

= Intertwining operator
{parent=Representation theory}
{wiki=Intertwining_operator}

An intertwining operator between representations $\rho$ and $\sigma$ is a linear map $T$ satisfying $T\rho(g)=\sigma(g)T$ for every group element $g$.

= Schur lemma
{c}
{parent=Representation theory}
{wiki=Schur%27s_lemma}

An intertwiner between irreducible complex representations is either zero or an isomorphism; an endomorphism of one irreducible is scalar.

= Proof of Schur lemma
{c}
{parent=Schur lemma}

The kernel and image of an intertwiner are invariant subspaces. Irreducibility therefore makes a nonzero intertwiner injective and surjective. For an endomorphism $T$ of a finite-dimensional irreducible complex representation, choose an eigenvalue $\lambda$. The noninvertible intertwiner $T-\lambda I$ must be zero, so $T=\lambda I$.

= Cyclic-center obstruction to a faithful irreducible representation
{parent=Schur lemma}

If a finite group has a faithful irreducible complex representation, every central element acts as a scalar by <Schur lemma>. Faithfulness embeds the centre into $\mathbb C^\times$, and every finite subgroup of $\mathbb C^\times$ is cyclic. Hence the group centre must be cyclic.

= Dual representation
{parent=Representation theory}
{wiki}

For a representation on $V$, the dual action on $V^*$ is $(g\varphi)(v)=\varphi(g^{-1}v)$.

= Character of a Hom representation
{parent=Dual representation}

For complex representations $V,W$ of a finite group,
$$
\operatorname{Hom}_{\mathbb C}(V,W)\cong V^*\otimes W.
$$
Under the $G\times G$ action $(g,h)\alpha=\sigma(h)\alpha\rho(g^{-1})$, its character is
$$
\chi_{\operatorname{Hom}(V,W)}(g,h)
=\overline{\chi_V(g)}\,\chi_W(h).
$$

= Two-sided regular representation decomposition
{parent=Character of a Hom representation}

Let $G\times G$ act on $\mathbb C G$ by $(g,h)x=gxh^{-1}$. Its character is zero unless $g$ and $h$ are conjugate, in which case it equals $|C_G(g)|$. Character column orthogonality gives the same character for
$$
\bigoplus_i\operatorname{Hom}_{\mathbb C}(V_i,V_i),
$$
where the $V_i$ run through the irreducible complex representations. Maschke's theorem therefore gives an isomorphism of the two representations.

= Invariant bilinear form as an intertwiner
{parent=Dual representation}

A bilinear form $B$ on $V$ is invariant exactly when $v\mapsto B(v,-)$ is a $G$-homomorphism $V\to V^*$. On an irreducible representation, Schur's lemma makes a nonzero such form <nondegenerate bilinear form>[nondegenerate] and unique up to scale.

= Symmetric-or-alternating dichotomy for invariant forms
{parent=Invariant bilinear form as an intertwiner}

Transposing a nonzero invariant bilinear form on an irreducible complex representation gives a scalar multiple $B^T=\lambda B$. Transposing twice yields $\lambda^2=1$, so the form is symmetric or alternating.

= Induced representation
{parent=Representation theory}
{wiki}

Induction extends a representation of a subgroup to the whole group; Frobenius reciprocity computes its multiplicities.

= Character formula for an induced representation
{parent=Induced representation}

For $H\leq G$ and an $H$-character $\chi$,
$$
(\operatorname{Ind}_H^G\chi)(g)=\frac1{|H|}
\sum_{\substack{x\in G\\x^{-1}gx\in H}}\chi(x^{-1}gx).
$$

= Induction from an abelian normal subgroup with a free character orbit
{parent=Induced representation}

Let $N\triangleleft G$ be abelian and let a linear character $\theta$ of $N$ have stabilizer exactly $N$ under conjugation by $G$. Then $\operatorname{Ind}_N^G\theta$ is irreducible of degree $[G:N]$. Its value is zero outside $N$, while on $n\in N$ it is the sum of the characters in the $G$-orbit of $\theta$.

= Irreducible characters of the affine semidirect product of orders eleven and five
{parent=Induction from an abelian normal subgroup with a free character orbit}

For $G=C_{11}\rtimes C_5$ with faithful action, the abelianization $G/C_{11}\cong C_5$ supplies five linear characters. The two orbits of size five on the nontrivial characters of $C_{11}$ induce two irreducible characters of degree five. Their squared degrees satisfy
$$
5\cdot1^2+2\cdot5^2=55,
$$
so these seven characters are all the irreducible characters of $G$.

= Tensor identity for induced representations
{parent=Induced representation}

For a $G$-representation $W$ and an $H$-representation $V$,
$$
\operatorname{Ind}_H^G(\operatorname{Res}_H^G W\otimes V)
\cong W\otimes\operatorname{Ind}_H^G V.
$$
On group-algebra tensors the isomorphism sends $x\otimes(w\otimes v)$ to $xw\otimes(x\otimes v)$.

= Frobenius reciprocity
{parent=Induced representation}
{c}
{wiki}

For $H\leq G$, an $H$-representation $V$, and a $G$-representation $W$,
$$
\operatorname{Hom}_G(\operatorname{Ind}_H^GV,W)
\cong
\operatorname{Hom}_H(V,\operatorname{Res}_H^GW).
$$
Equivalently, induction and restriction are adjoint for character inner products.

= Mackey restriction formula
{parent=Induced representation}
{c}
{wiki=Mackey_theory}

For $H,K\leq G$ and a $K$-representation $V$,
$$
\operatorname{Res}_H^G\operatorname{Ind}_K^GV
\cong
\bigoplus_{x\in H\backslash G/K}
\operatorname{Ind}_{H\cap xKx^{-1}}^H
\operatorname{Res}_{H\cap xKx^{-1}}^{xKx^{-1}}({}^xV),
$$
where ${}^xV(xkx^{-1})=V(k)$.

= Mackey irreducibility criterion
{c}
{parent=Mackey restriction formula}
{wiki=Mackey_theory#Mackey%27s_irreducibility_criterion}

For an irreducible $H$-representation $V$, the induced representation $\operatorname{Ind}_H^GV$ is irreducible exactly when, for every nonidentity double coset representative $x\in H\backslash G/H$, the restrictions of $V$ and its $x$-conjugate to $H\cap xHx^{-1}$ have inner product zero.

= Bruhat decomposition of SL2 over a finite field
{parent=Mackey restriction formula}
{c}
{wiki=Bruhat_decomposition}

For the upper-triangular subgroup $B$ of $SL_2(\mathbb F_p)$ and
$$
w=\begin{pmatrix}0&-1\\1&0\end{pmatrix},
$$
the two double cosets are $B$ and $BwB$, and $B\cap wBw^{-1}=T$, the diagonal subgroup.

= Mackey inner product for the finite SL2 principal series
{parent=Bruhat decomposition of SL2 over a finite field}

For one-dimensional representations $\theta,\varphi$ of the upper-triangular subgroup $B\leq SL_2(\mathbb F_p)$,
$$
\left\langle\operatorname{Ind}_B^G\theta,
\operatorname{Ind}_B^G\varphi\right\rangle_G
=\mathbf1_{\theta=\varphi}
+\mathbf1_{\theta^w|_T=\varphi|_T},
$$
where $\theta^w(t)=\theta(w^{-1}tw)$.

= Irreducible principal series of finite SL2
{parent=Mackey inner product for the finite SL2 principal series}

Every linear character of the upper-triangular subgroup $B\leq SL_2(\mathbb F_q)$ with $q\geq4$ is obtained from a multiplicative character $\theta$ of $\mathbb F_q^\times$. Its induction to $SL_2(\mathbb F_q)$ is irreducible exactly when $\theta^2\ne1$.

= Square orbits of additive characters of a finite field
{parent=Induced representation}

The diagonal subgroup of $SL_2(\mathbb F_p)$ conjugates $u(x)$ to $u(a^2x)$. It therefore has two orbits, each of size $(p-1)/2$, on the nontrivial characters $\chi(v\mathbin\cdot)$ of the unipotent subgroup: one indexed by squares and one by nonsquares.

= Induction from the diagonal subgroup of upper-triangular SL2
{parent=Square orbits of additive characters of a finite field}

If $B=U\rtimes T$ is the upper-triangular subgroup of $SL_2(\mathbb F_p)$ and $\theta$ is one-dimensional on $T$, then
$$
\operatorname{Ind}_T^B\theta
$$
is the direct sum of three pairwise nonisomorphic irreducibles. Their restrictions to $U$ have character supports given by the trivial character, the square orbit, and the nonsquare orbit, so their dimensions are $1,(p-1)/2,(p-1)/2$.

= Irreducible representation
{parent=Representation theory}
{wiki}

An irreducible representation has no nonzero proper invariant subspace.

= Eigenvalue orbit under conjugation
{parent=Irreducible representation}

If conjugation sends an operator to its inverse, the conjugating element sends its $\lambda$-eigenspace to the $\lambda^{-1}$-eigenspace.

= Irreducible complex representations of a finite dihedral group
{parent=Irreducible representation}

For $D_{2n}=\langle r,s:r^n=s^2=1,\ srs=r^{-1}\rangle$, every irreducible complex representation has degree at most two. If $v$ is an eigenvector of $r$ with eigenvalue $\lambda$, then $sv$ has eigenvalue $\lambda^{-1}$, so irreducibility makes the representation equal to $\operatorname{span}\{v,sv\}$.

= Two-dimensional representations of a finite dihedral group
{parent=Irreducible complex representations of a finite dihedral group}

For $\omega=e^{2\pi i/n}$ and $1\leq j<n/2$, define
$$
r\longmapsto
\begin{pmatrix}\omega^j&0\\0&\omega^{-j}\end{pmatrix},
\qquad
s\longmapsto
\begin{pmatrix}0&1\\1&0\end{pmatrix}.
$$
These representations are irreducible and pairwise nonisomorphic. They give all two-dimensional irreducibles: $(n-1)/2$ of them for odd $n$, and $n/2-1$ for even $n$.

= Faithful irreducible representation of D8
{c}
{parent=Two-dimensional representations of a finite dihedral group}

For $D_8=\langle r,s:r^4=s^2=1,\ srs=r^{-1}\rangle$, the matrices
$$
r\longmapsto
\begin{pmatrix}i&0\\0&-i\end{pmatrix},
\qquad
s\longmapsto
\begin{pmatrix}0&1\\1&0\end{pmatrix}
$$
define a faithful irreducible two-dimensional complex representation.

= Infinite dihedral group
{parent=Representation theory}
{wiki}

The infinite dihedral group is $\langle r,t\mid t^2=1,\ trt^{-1}=r^{-1}\rangle$.

= Representation of the infinite dihedral group
{parent=Infinite dihedral group}

Its irreducible complex representations have dimension at most two; the two-dimensional family pairs eigenvalues $\lambda$ and $\lambda^{-1}$ of the rotation generator.

= Induced two-dimensional dihedral representation
{parent=Representation of the infinite dihedral group}

For $\lambda\ne\pm1$, the rotation acts by $\operatorname{diag}(\lambda,\lambda^{-1})$ and the reflection interchanges the two eigenlines.

= One-dimensional character
{parent=Representation theory}
{wiki=Character_theory}

A one-dimensional character is a homomorphism from a group to the multiplicative group of its scalar field.

= One-dimensional characters factor through the abelianization
{parent=One-dimensional character}

Every <one-dimensional character> kills the <commutator subgroup> and therefore factors through the <abelianization>. Conversely, every character of the abelianization pulls back to a one-dimensional character of the group.

= Finite quotient representation
{parent=Representation theory}

A representation factors through a finite quotient exactly when its kernel contains the defining kernel of that quotient.

= Indecomposable representation
{parent=Representation theory}
{wiki=Indecomposable_module}

An indecomposable representation cannot be expressed as a direct sum of two nonzero invariant subspaces.

= Nonsplit extension of representations
{parent=Indecomposable representation}

A nonsplit extension contains an invariant subrepresentation with no invariant complementary subspace.

= Unipotent representation
{parent=Indecomposable representation}

A unipotent operator has every eigenvalue equal to one and may create a nonsplit Jordan-block extension.

= Coding theory
{parent=Algebra}
{wiki}

= Binary block code
{parent=Coding theory}
{wiki=Block_code}

A binary $[n,m,d]$ code is a set of $m$ distinct words in $\mathbb F_2^n$ whose minimum pairwise <Hamming distance> is $d$. Here $n$ is the block length and $m$ is the number of codewords.

= Hamming distance
{title2=$d_H$}
{parent=Binary block code}
{c}
{wiki}

The Hamming distance between two equal-length words is the number of coordinates in which they differ.

= One-time pad
{parent=Coding theory}
{wiki}

For messages in a <finite additive group>, a one-time pad encrypts $X$ as $C=X+K$, where the key $K$ is uniform and <independent random variables>[independent] of $X$, as long as the message, and never reused. Then $C$ is uniform and <independent random variables>[independent] of $X$, giving <perfect secrecy>.

= Perfect secrecy
{parent=One-time pad}
{wiki}

An encryption scheme has perfect secrecy when the ciphertext and plaintext are <independent random variables>, so observing the ciphertext does not change the plaintext's <probability distribution>.

= Malleability of an additive one-time pad
{parent=One-time pad}

If an interceptor knows that an additive one-time-pad ciphertext $C$ encrypts plaintext $P$ and wants it to decrypt as $P'$, replacing it by
$$
C'=C-P+P'
$$
works without knowing the key, since $C'=P'+K$.

= Discrete memoryless channel
{parent=Coding theory}
{wiki=Discrete_memoryless_channel}

A discrete memoryless channel has finite input and output alphabets and transition probabilities $P(y\mid x)$. Conditional on the input symbols, different channel uses have independent outputs governed by the same transition probabilities.

= Shannon second coding theorem
{parent=Coding theory}
{c}
{wiki=Noisy-channel_coding_theorem}

For a discrete memoryless channel, operational capacity equals
$$
C=\max_{P_X}I(X;Y).
$$
Rates below $C$ admit codes whose error tends to zero, while rates above $C$ cannot have vanishing error.

= Linear-feedback shift register
{parent=Coding theory}
{wiki=Linear-feedback_shift_register}

A binary linear-feedback shift register of degree $d$ updates a $d$-bit state by shifting and inserting a fixed linear combination over $\mathbb F_2$. With recurrence
$$
s_{n+d}=a_{d-1}s_{n+d-1}+\cdots+a_0s_n,
$$
its feedback polynomial is $x^d+a_{d-1}x^{d-1}+\cdots+a_0$.

= LFSR
{synonym}

= Feedback shift register
{parent=Linear-feedback shift register}
{wiki=Feedback_shift_register}

A feedback shift register updates a finite state by shifting its stored symbols and inserting the value of a feedback function of the previous state. The feedback function need not be linear.

= Berlekamp-Massey algorithm
{c}
{parent=Linear-feedback shift register}
{wiki=Berlekamp%E2%80%93Massey_algorithm}

The Berlekamp-Massey algorithm finds the shortest linear recurrence satisfied by a finite sequence over a field. It processes symbols incrementally, updating the connection polynomial whenever the current recurrence has a nonzero discrepancy.

= Minimal recurrence for the binary prefix 11001011
{parent=Berlekamp-Massey algorithm}

The shortest binary linear recurrence producing the prefix $11001011$ has length three:
$$
x_n=x_{n-2}+x_{n-3}.
$$
Its connection polynomial is $1+D^2+D^3$.

= Period bound for a linear-feedback shift register
{parent=Linear-feedback shift register}

A degree-$d$ binary register has $2^d$ states. The zero state is fixed, so a nonzero periodic orbit has length at most $2^d-1$.

= Feedback-polynomial parity condition for maximal period
{parent=Linear-feedback shift register}

A maximal-period feedback polynomial cannot have $1$ as a root. Over $\mathbb F_2$, this says
$$
1+a_{d-1}+\cdots+a_0\ne0,
$$
so an even number of the coefficients $a_i$ equal one and the full monic polynomial has an odd number of nonzero coefficients.

= Zero-run lower bound for a linear-feedback shift register
{parent=Linear-feedback shift register}

If an output prefix contains $d$ consecutive zeros followed by a one, no register of degree at most $d$ can generate it: the all-zero state would have been reached and would remain zero forever.

= Minimal linear-feedback shift register for the prefix 100000001
{parent=Zero-run lower bound for a linear-feedback shift register}

The seven consecutive zeros followed by one force degree at least eight. Degree eight is attained by $s_{n+8}=s_n$, with initial state $10000000$ and feedback polynomial $x^8+1$.

= Trace-generated maximal-period linear-feedback sequence
{parent=Linear-feedback shift register}

Let $K=\mathbb F_{2^d}$, let $\alpha$ generate $K^\times$, and let $T:K\to\mathbb F_2$ be nonzero linear with <nondegenerate bilinear form> $(x,y)\mapsto T(xy)$. If the <minimal polynomial> of $\alpha$ is $P(X)=X^d+\sum_{j<d}c_jX^j$, then $x_n=T(\alpha^n)$ satisfies
$$
x_{n+d}=\sum_{j<d}c_jx_{n+j},
$$
so it is produced by an LFSR of length at most $d$. Its period is exactly $2^d-1$: any period $r$ would imply $T((\alpha^r-1)y)=0$ for every $y\in K$, hence $\alpha^r=1$ by nondegeneracy.

= Linear code
{parent=Coding theory}
{wiki}

A binary linear $[n,k,d]$ code is a $k$-dimensional subspace of $\mathbb F_2^n$ with least nonzero weight $d$. A parity-check matrix presents it as a kernel.

= Even-weight binary code
{parent=Linear code}
{wiki=Parity_bit}

The length-$n$ even-weight code consists of binary words whose coordinates sum to zero. It has parameters $[n,n-1,2]$.

= Binary repetition code
{parent=Linear code}
{wiki=Repetition_code}

The binary repetition code is $\{0^n,1^n\}$ and has parameters $[n,1,n]$.

= Zero code
{parent=Linear code}

The zero code contains only the zero word and has dimension zero.

= Whole-space linear code
{parent=Linear code}

The whole-space binary code is $\mathbb F_2^n$, with parameters $[n,n,1]$.

= Hamming weight
{title2=$\operatorname{wt}$}
{parent=Linear code}
{c}
{wiki}

The Hamming weight $\operatorname{wt}(x)$ of a word is the number of its nonzero coordinates.

= Minimum Hamming distance of a linear code
{title2=$d$}
{parent=Linear code}

The minimum Hamming distance of a nonzero <linear code> is the smallest <Hamming weight> of a nonzero codeword.

= Generator matrix
{title2=$G$}
{parent=Linear code}
{wiki}

A generator matrix for a <linear code> has a basis of the code as its rows, so its row space is the code.

= Parity-check matrix
{title2=$H$}
{parent=Linear code}
{wiki}

A parity-check matrix $H$ for a <linear code> $C$ satisfies
$$
C=\ker H=\{c:Hc^T=0\}.
$$

= Parity-check extension of a linear code
{parent=Linear code}

The parity-check extension of a binary <linear code> appends to each codeword the sum of its coordinates in $\mathbb F_2$, making every extended word have even <Hamming weight>.

= Punctured code
{parent=Linear code}
{wiki=Puncturing_(coding_theory)}

Puncturing a code at one coordinate deletes that coordinate from every codeword. For a code of minimum distance $d\geq2$, puncturing preserves its number of words and gives minimum distance $d$ or $d-1$.

= Shortened code
{parent=Linear code}
{wiki=Puncturing_(coding_theory)}

Shortening a code at one coordinate first retains only words with a chosen symbol there and then deletes that coordinate. The minimum distance cannot decrease. In a binary code one coordinate-symbol fibre contains at least half of the words.

= Dual code
{title2=$C^\perp$}
{parent=Linear code}
{wiki}

The dual code $C^\perp$ consists of the words orthogonal to every word of $C$ under the standard dot product. A <parity-check matrix> for $C$ is a <generator matrix> for $C^\perp$.

= Reed-Muller code
{c}
{parent=Linear code}
{wiki=Reed%E2%80%93Muller_code}

The binary Reed-Muller code $\operatorname{RM}(d,r)$ evaluates Boolean polynomials of degree at most $r$ on $\mathbb F_2^d$. It has parameters
$$
\left[2^d,\sum_{j=0}^r\binom dj,2^{d-r}\right].
$$
The dimension counts square-free monomials, and the minimum weight follows inductively from the decomposition $(u,u+v)$.

= Bar product of binary linear codes
{title2=$C_1|C_2$}
{parent=Reed-Muller code}

For length-$n$ binary linear codes $C_2\subseteq C_1$, their bar product is
$$
C_1|C_2=\{(u,u+v):u\in C_1,\ v\in C_2\}.
$$
It has length $2n$ and dimension $\dim C_1+\dim C_2$.

= Reed-Muller bar-product recursion
{c}
{parent=Bar product of binary linear codes}

For $0<r<d$,
$$
\operatorname{RM}(d,r)
=\operatorname{RM}(d-1,r)
\mid\operatorname{RM}(d-1,r-1).
$$
Consequently
$$
\dim\operatorname{RM}(d,r)=\sum_{j=0}^r\binom dj.
$$

= Dual of a Reed-Muller code
{c}
{parent=Reed-Muller code}

Under the standard binary inner product,
$$
\operatorname{RM}(d,r)^\perp
=\operatorname{RM}(d,d-r-1).
$$
The product of representing polynomials has degree at most $d-1$, so its evaluation word has even weight; equality follows by comparing dimensions.

= Hamming code
{c}
{parent=Linear code}
{wiki}

The binary Hamming code of redundancy $r$ has parameters $[2^r-1,2^r-r-1,3]$. Its parity-check matrix has every nonzero vector of $\mathbb F_2^r$ as a column.

= Binary simplex code
{parent=Hamming code}
{wiki=Simplex_code}

The binary simplex code is the dual of the binary Hamming code. For redundancy $r$ it has parameters $[2^r-1,r,2^{r-1}]$.

= Perfectness of a Hamming code
{parent=Hamming code}

A Hamming code has $2^{2^r-r-1}$ words, and each radius-one Hamming ball contains $1+(2^r-1)=2^r$ vectors. Its minimum distance is three, so these balls are disjoint, and their total size is $2^{2^r-1}$. They therefore partition the ambient space, making the code perfect.

= Minimum-distance error-detection and correction guarantee
{parent=Linear code}

A code of minimum Hamming distance $d$ detects every pattern of at most $d-1$ errors and uniquely corrects every pattern of at most $\lfloor(d-1)/2\rfloor$ errors. The correction claim follows because Hamming balls of that radius around distinct codewords are disjoint.

= Cyclic code
{parent=Coding theory}
{wiki}

A length-$n$ cyclic code over $\mathbb F_q$ is an ideal of $\mathbb F_q[X]/(X^n-1)$. It is generated by a monic divisor of $X^n-1$.

= Generator polynomial of a cyclic code
{title2=$g(X)$}
{parent=Cyclic code}
{wiki=Cyclic_code#Definition}

The unique monic divisor $g(X)$ of $X^n-1$ generating a cyclic code is its generator polynomial. The code has dimension $n-\deg g$.

= Dual of a cyclic code
{parent=Cyclic code}

The dual of a cyclic code is cyclic. If $g(X)h(X)=X^n-1$, then the dual generator is the monic reciprocal polynomial $h^*(X)$.

= Binary cyclic codes of length seven
{parent=Cyclic code}

Over $\mathbb F_2$,
$$
X^7-1=(X+1)(X^3+X+1)(X^3+X^2+1).
$$
Its eight monic divisors generate the eight cyclic codes of length seven: the whole-space, even-weight, two Hamming, two simplex, repetition, and zero codes.

= BCH code
{parent=Cyclic code}
{c}
{wiki=BCH_code}

Given a primitive $n$th root $\alpha$, a BCH code of design distance $\delta$ has a generator polynomial vanishing at $\delta-1$ consecutive powers $\alpha^b,\ldots,\alpha^{b+\delta-2}$.

= Error locator polynomial
{title2=$\sigma(Z)$}
{parent=BCH code}
{wiki}

For error locations $i_1,\ldots,i_\nu$ in a cyclic code with primitive root $\alpha$, the error locator polynomial is $\sigma(Z)=\prod_h(1-\alpha^{i_h}Z)$. Its roots encode the error positions.

= BCH bound
{parent=BCH code}
{c}

The minimum distance of a BCH code is at least its design distance. A hypothetical word of smaller weight yields a square Vandermonde system from the consecutive-root equations; its support elements are distinct, so the determinant is nonzero and every word coefficient must vanish.

= Parity extension
{parent=Coding theory}

Appending a parity bit makes every word even-weight. A binary $[n,m,d]$ code becomes an $[n+1,m,d^+]$ code, where
$$
d^+=
\begin{cases}
d,&d\text{ even},\\
d+1,&d\text{ odd}.
\end{cases}
$$

= Hamming bound
{c}
{parent=Coding theory}
{wiki}

A binary length-$n$ code of size $M$ and distance at least $2t+1$ satisfies
$$
M\sum_{j=0}^t\binom nj\le2^n.
$$
Equality means the code is perfect.

= Rabin cryptosystem
{c}
{parent=Coding theory}
{wiki=Rabin_cryptosystem}

Rabin encryption sends $m$ to $m^2$ modulo a product of two secret primes. Decryption computes the four square roots by the Chinese remainder theorem.

= Discrete logarithm problem
{parent=Coding theory}
{wiki=Discrete_logarithm}

Given a cyclic group generator $g$ and an element $h$, the discrete logarithm problem asks for an exponent $a$ such that $g^a=h$.

= Diffie-Hellman key exchange
{c}
{parent=Coding theory}
{wiki=Diffie%E2%80%93Hellman_key_exchange}

Alice publishes $g^a$ and Bob publishes $g^b$ in a public cyclic group; each can then compute the shared secret $g^{ab}$. Its security relies on the difficulty of recovering the exponents or otherwise solving the computational Diffie--Hellman problem.

= Decipherable code
{parent=Coding theory}
{wiki=Unique_decodability}

A code is decipherable, or uniquely decodable, when every finite concatenation of codewords has at most one decomposition into source codewords.

= Uniquely decodable code
{synonym}

= Prefix code
{parent=Decipherable code}
{wiki=Prefix_code}

In a prefix code no codeword is the prefix of another, so concatenated messages have unique instantaneous decoding.

= Kraft inequality
{parent=Prefix code}
{c}
{wiki=Kraft–McMillan_inequality}

The lengths of every decipherable binary code satisfy
$$
\sum_i2^{-l_i}\leq1.
$$
Conversely, any positive integer lengths satisfying this bound can be realized by a <prefix code>.

= Total length lower bound for a decipherable binary code
{parent=Kraft inequality}

If a decipherable binary code has $N$ codewords of lengths $l_1,\ldots,l_N$, then
$$
\sum_{i=1}^N l_i\geq N\log_2N.
$$
This follows by applying the entropy lower bound with the uniform source distribution.

= Shannon-Fano coding
{parent=Prefix code}
{c}
{wiki=Shannon–Fano_coding}

Shannon--Fano coding assigns a symbol of probability $p_i$ a prefix-code length near $-\log_2p_i$.

= Cumulative Shannon code
{c}
{parent=Prefix code}
{wiki=Shannon_coding}

Order source probabilities as $p_1\geq\cdots\geq p_N$, put
$$
b_i=\sum_{j<i}p_j,
\qquad
l_i=\lceil-\log_2p_i\rceil,
$$
and take the first $l_i$ binary digits of $b_i$ as the codeword for symbol $i$. Since $b_j-b_i\geq p_i\geq2^{-l_i}$ for $j>i$, two cumulative probabilities cannot lie in the same dyadic interval selected by the earlier codeword. The resulting code is prefix-free.

= Entropy lower bound for prefix codes
{parent=Prefix code}

Every binary prefix code has expected length at least its source entropy; Shannon lengths achieve expected length strictly below $H+1$.

= Binary symmetric channel
{parent=Coding theory}
{wiki}

A binary symmetric channel independently flips each transmitted bit with a fixed crossover probability $p$.

= Undetected error
{parent=Binary symmetric channel}
{wiki=Error_detection_and_correction}

An error is undetected by codeword-membership checking when the received word is a different valid codeword. For a linear code this occurs exactly when the nonzero channel-error vector is itself a codeword.

= Completely noisy binary symmetric channel
{parent=Binary symmetric channel}

At crossover probability $p=1/2$, the two rows of the <transition matrix> of a <binary symmetric channel> coincide. Its output is then <independent random variables>[independent] of its input and its <binary symmetric channel capacity>[capacity] is zero.

= Positive-rate coding bound below one-quarter crossover
{parent=Binary symmetric channel}

If $p<1/4$, choose $p<\delta<1/4$. Binary codes of relative minimum distance above $2\delta$ have positive asymptotic rate by a greedy packing bound, and nearest-codeword decoding fails only when more than $\delta n$ channel errors occur. That probability tends to zero.

= Maximum a posteriori decoding
{parent=Coding theory}
{wiki=Maximum_a_posteriori_estimation}

Maximum a posteriori decoding chooses the codeword maximizing prior probability times channel likelihood and minimizes average decision error.

= Maximum-likelihood decoding
{parent=Coding theory}
{wiki}

Maximum-likelihood decoding chooses the codeword under which the received word has greatest conditional probability.

= Minimum-distance decoding
{parent=Coding theory}
{wiki}

Minimum-distance decoding chooses a codeword closest to the received word; on a binary symmetric channel with $p<1/2$, it is maximum-likelihood decoding under Hamming distance.

= Linear algebra
{parent=Algebra}
{wiki}

= Cartesian basis
{parent=Linear algebra}
{c}
{wiki=Cartesian_coordinate_system}

A Cartesian basis is an orthonormal basis associated with perpendicular Cartesian coordinate axes.

= Bilinear form
{parent=Linear algebra}
{wiki}

A bilinear form on a vector space $V$ over a field $F$ is a map $B:V\times V\to F$ that is linear in each argument separately.

= Radical of a bilinear form
{parent=Bilinear form}
{wiki=Bilinear_form\#Definitions}

The radical of a symmetric bilinear form $B$ is
$$
\operatorname{rad}B=\{v:B(v,w)=0\text{ for every }w\}.
$$
The form is nondegenerate exactly when its radical is zero.

= Nondegenerate bilinear form
{parent=Bilinear form}
{wiki=Bilinear_form\#Non-degenerate_bilinear_forms}

A <bilinear form> $B$ on a finite-dimensional <vector space> is nondegenerate when
$$
B(v,w)=0\text{ for every }v\quad\Longrightarrow\quad w=0,
$$
equivalently when $w\mapsto B(-,w)$ is an isomorphism from $V$ to its <dual space>.

= Symplectic vector space
{parent=Nondegenerate bilinear form}
{wiki}

A symplectic vector space is a vector space equipped with a nondegenerate antisymmetric bilinear form. Every finite-dimensional symplectic vector space has even dimension: splitting off one symplectic two-plane leaves a smaller nondegenerate antisymmetric space, and induction completes the decomposition.

= Representation of a bilinear form relative to a nondegenerate bilinear form
{parent=Nondegenerate bilinear form}

If $B_1$ is a <nondegenerate bilinear form> on a finite-dimensional vector space and $B_2$ is any bilinear form, there is a unique <linear map> $\alpha$ such that
$$
B_2(v,w)=B_1(v,\alpha w).
$$
The right radical of $B_2$ is $\ker\alpha$.

= Inertia of a bilinear form
{title2=$(p,q,r)$}
{parent=Bilinear form}
{wiki=Sylvester%27s_law_of_inertia}

The inertia of a real symmetric bilinear form is the triple giving the numbers of positive, negative, and zero squares in a diagonalization. <Sylvester's law of inertia> makes it independent of the chosen basis.

= Sesquilinear form
{parent=Linear algebra}
{wiki}

A complex sesquilinear form is linear in one argument and conjugate-linear in the other. Here the first argument is taken to be linear.

= Hermitian form
{parent=Sesquilinear form}
{wiki=Sesquilinear_form}

A Hermitian form is a complex sesquilinear form $H$ satisfying $H(w,v)=\overline{H(v,w)}$.

= Matrix of a Hermitian form
{parent=Hermitian form}

For a basis $(v_i)$, the matrix $A$ of a <Hermitian form> has entries $A_{ij}=H(v_i,v_j)$. With the convention that $H$ is linear in its first argument, coordinate columns $x,y$ satisfy $H(x,y)=x^TA\overline y$.

= Vector space
{parent=Linear algebra}
{wiki}

A vector space consists of <vectors> that may be added and multiplied by <scalars>, subject to the vector-space axioms.

= Linear combination
{parent=Vector space}
{wiki}

A linear combination of vectors $v_1,\ldots,v_n$ is a sum $a_1v_1+\cdots+a_nv_n$ with scalar coefficients $a_i$.

= Linear combinations
{synonym}

= Scalar multiple
{parent=Vector space}
{wiki=Scalar_multiplication}

A scalar multiple of a vector $v$ is a vector of the form $av$ for a scalar $a$.

= Dimension of a vector space
{title2=$\dim V$}
{parent=Vector space}
{wiki=Dimension_(vector_space)}

The dimension of a vector space is the cardinality of any of its <bases>.

= Direct sum
{title2=$\oplus$}
{parent=Vector space}
{wiki}

The direct sum $U\oplus W$ of two <vector space>[vector spaces] consists of pairs $(u,w)$ with componentwise operations. A vector space is an internal direct sum of subspaces $U$ and $W$ exactly when every vector has a unique expression $u+w$ with $u\in U$ and $w\in W$.

= Dimension formula for a sum of subspaces
{parent=Direct sum}

For finite-dimensional subspaces $U,W$ of a common vector space,
$$
\dim(U+W)+\dim(U\cap W)=\dim U+\dim W.
$$

= Direct-sum complement
{parent=Direct sum}

For every subspace $U$ of a finite-dimensional vector space $V$, extending a basis of $U$ to a basis of $V$ produces a subspace $W$ such that $V=U\oplus W$.

= Linear independence
{parent=Vector space}
{wiki}

Vectors $v_1,\ldots,v_n$ are linearly independent when
$$
a_1v_1+\cdots+a_nv_n=0
$$
implies $a_1=\cdots=a_n=0$.

= Linearly independent vectors
{synonym}

= Linear dependence
{parent=Linear independence}
{wiki}

Vectors are linearly dependent when some nonzero choice of coefficients gives a vanishing linear combination.

= Basis
{parent=Vector space}
{wiki=Basis_(linear_algebra)}

A basis is a <linear independence>[linearly independent] spanning family of <vectors> in a <vector space>.

= Bases
{synonym}

= Vector
{parent=Vector space}
{wiki=Euclidean_vector}

A vector is an element of a <vector space>.

= Cross product
{title2=$\times$}
{parent=Vector}
{wiki}

The cross product $a\times b$ of two three-dimensional <vectors> is perpendicular to both, has magnitude $|a||b|\sin\theta$, and is oriented by the right-hand rule.

= Scalar
{parent=Vector space}
{wiki=Scalar_(mathematics)}

A scalar is an element of the field over which a <vector space> is defined.

= Linear map
{parent=Vector space}
{wiki=Linear_map}

A linear map preserves vector addition and scalar multiplication.

= First isomorphism theorem for vector spaces
{parent=Linear map}
{c}
{wiki=Isomorphism_theorems#Modules}

For a linear map $T:V\to W$, the induced map
$$
V/\ker T\longrightarrow\operatorname{im}T,
\qquad
v+\ker T\longmapsto T(v),
$$
is a vector-space isomorphism.

= Linear isomorphism
{parent=Linear map}
{wiki=Linear_map\#Isomorphism}

A linear isomorphism is a <bijection>[bijective] <linear map>. Its inverse is also linear.

= Linear function
{parent=Linear map}
{wiki=Linear_function}

A real linear function on $\mathbb R^n$ has the form $x\mapsto c^Tx$.

= Linearity
{parent=Linear map}
{wiki=Linear_map}

Linearity is the property $T(ax+by)=aT(x)+bT(y)$.

= Linear
{synonym}

= Superposition principle
{parent=Linear map}
{wiki}

For a homogeneous <linear> equation, every linear combination of solutions is again a solution.

= Linear operator
{parent=Linear map}
{wiki=Linear_map}

A linear operator is a <linear map> from a <vector space> to itself.

= Unitary operator
{parent=Linear operator}
{wiki}

A unitary operator $U$ on a complex vector space with an <inner product> satisfies $U^*U=UU^*=I$ and therefore preserves inner products.

= Unitary operators
{synonym}

= Conjugate linear operators
{parent=Linear operator}
{wiki=Similar_matrix}

Linear operators $\alpha$ and $\beta$ on the same finite-dimensional vector space are conjugate when $\beta=s^{-1}\alpha s$ for some linear isomorphism $s$. Their matrices in any fixed basis are similar.

= Conjugation operator on an endomorphism space
{parent=Conjugate linear operators}

For an invertible operator $\beta$ on $V$, the map
$$
\phi_\beta:\operatorname{End}(V)\to\operatorname{End}(V),
\qquad A\mapsto\beta^{-1}A\beta
$$
is a linear isomorphism. Conjugate choices of $\beta$ induce conjugate operators $\phi_\beta$.

= Matrix
{parent=Linear map}
{wiki=Matrix_(mathematics)}

A matrix represents a <linear map> after bases have been chosen for its domain and codomain.

= Matrix power
{title2=$A^k$}
{parent=Matrix}

For a square <matrix> $A$ and a <integer>[nonnegative integer] $k$, the matrix power is the repeated product
$$
A^k=\underbrace{AA\cdots A}_{k\text{ factors}},
$$
with $A^0=I$.

= Matrix element
{title2=$A_{ij}$}
{parent=Matrix}
{wiki=Matrix_(mathematics)}

A matrix element $A_{ij}$ is the entry in row $i$ and column $j$ of a <matrix> $A$.

= Matrix unit
{title2=$E_{ij}$}
{parent=Matrix}
{wiki}

The matrix unit $E_{ij}$ has entry one in row $i$, column $j$, and zero in every other position. The matrix units form the standard <basis> of the <vector space> of matrices of a fixed size.

= Matrix rank
{title2=$\operatorname{rank}A$}
{parent=Matrix}
{wiki=Rank_(linear_algebra)}

The rank of a matrix is the <dimension of a vector space>[dimension] of its column space, equivalently the dimension of its row space.

= Identity matrix
{title2=$I$}
{parent=Matrix}
{wiki}

The identity matrix has ones on its main diagonal and zeros elsewhere, and satisfies $IA=AI=A$ whenever the products are defined.

= Matrices
{synonym}

= Rank-one matrix
{parent=Matrix}
{wiki=Rank_(linear_algebra)#Rank_one}

A nonzero matrix has rank one exactly when it can be written as an outer product $uv^T$.

= Upper bidiagonal matrix
{parent=Matrix}
{wiki=Bidiagonal_matrix}

An upper bidiagonal matrix can have nonzero entries only on its main diagonal and first superdiagonal.

= Matrix multiplication
{parent=Matrix}
{wiki}

Matrix multiplication represents composition of linear maps. Its entries are $(AB)_{ij}=\sum_kA_{ik}B_{kj}$.

= Outer product
{parent=Matrix multiplication}
{wiki}

The outer product of column vectors $u$ and $v$ is the rank-at-most-one matrix $uv^T$, whose $(i,j)$ entry is $u_iv_j$.

= Upper Hessenberg matrix
{parent=Matrix}
{c}
{wiki=Hessenberg_matrix}

An upper Hessenberg matrix has zero entries below its first subdiagonal: $a_{ij}=0$ whenever $i>j+1$.

= Permutation matrix
{title2=$P$}
{parent=Matrix}
{wiki}

A permutation matrix has exactly one entry equal to one in each row and column and zeros elsewhere. Left or right multiplication permutes coordinates, and $P^{-1}=P^T$.

= Matrix representation of a linear map
{parent=Matrix}

Given ordered bases $\mathcal B=(v_j)$ and $\mathcal C=(w_i)$, the matrix $[T]_{\mathcal C\leftarrow\mathcal B}$ has column $j$ equal to the $\mathcal C$-coordinate vector of $T(v_j)$.

= Kronecker product
{title2=$\otimes$}
{parent=Matrix}
{wiki=Kronecker_product}

For matrices $A$ and $B$, the Kronecker product $A\otimes B$ replaces each entry $a_{ij}$ by the block $a_{ij}B$. It satisfies
$$
(A\otimes B)(C\otimes D)=AC\otimes BD
$$
whenever the products are defined.

= Singular value decomposition
{parent=Linear algebra}
{wiki}

Every complex matrix has a factorization $A=V\Sigma U^\dagger$ with $U,V$ unitary and $\Sigma$ diagonal with nonnegative entries. The squared singular values are the eigenvalues of $A^\dagger A$.

= Singular value
{parent=Singular value decomposition}
{wiki}

The singular values of a <matrix> $A$ are the nonnegative <square root>[square roots] of the <eigenvalues> of $A^*A$.

= Unimodular matrix
{parent=Linear algebra}
{wiki}

An integral square matrix is unimodular when its determinant is $1$ or $-1$. Its inverse again has integral entries, so it defines an invertible change of integer coordinates.

= Householder reflection
{c}
{parent=Linear algebra}
{wiki=Householder_transformation}

For unit $v$, $I-2vv^T$ is an orthogonal reflection. Successive Householder similarities reduce a symmetric matrix while preserving symmetry.

= Change of basis
{parent=Linear algebra}
{wiki}

= Circulant matrix
{parent=Linear algebra}
{wiki}

= Invertible matrix
{parent=Linear algebra}
{wiki}

= Kernel of a linear map
{parent=Linear algebra}
{wiki}

= Kernel
{synonym}

= Rank-nullity theorem
{parent=Kernel of a linear map}
{wiki}

For a linear map $T:V\to W$ with finite-dimensional domain,
$$
\dim V=\dim\ker T+\dim\operatorname{im}T.
$$
If a basis of $\ker T$ is extended to a basis of $V$, the images of the added basis vectors form a basis of $\operatorname{im}T$. Counting the two parts proves the formula.

= Dual space
{parent=Linear algebra}
{wiki}

The dual space of a vector space $V$ over $F$ is $V^*=\operatorname{Hom}_F(V,F)$. If $V$ is finite-dimensional, then $\dim V^*=\dim V$.

= Linear functional
{parent=Dual space}
{wiki}

A linear functional is a linear map from a vector space to its scalar field.

= Linear functionals
{synonym}

= Dual vector space
{synonym}

= Dual basis
{parent=Dual space}
{wiki}

For a basis $e_1,\ldots,e_n$ of $V$, its dual basis $e_1^*,\ldots,e_n^*$ is characterized by $e_i^*(e_j)=\delta_{ij}$.

= Annihilator of a vector subspace
{parent=Dual space}
{wiki=Annihilator_(ring_theory)\#Linear_algebra}

For $U\leq V$,
$$
U^\circ=\{f\in V^*:f|_U=0\}.
$$
In finite dimensions, $\dim U^\circ=\dim V-\dim U$.

= Dual map
{parent=Dual space}
{wiki=Transpose_of_a_linear_map}

The dual of $\alpha:V\to W$ is $\alpha^*:W^*\to V^*$ defined by $\alpha^*(g)=g\circ\alpha$. In finite dimensions,
$$
\ker\alpha^*=(\operatorname{im}\alpha)^\circ,
\qquad
\operatorname{im}\alpha^*=(\ker\alpha)^\circ.
$$

= Equality of row rank and column rank
{parent=Dual map}
{wiki=Rank_(linear_algebra)\#Column_rank_=_row_rank}

For the map represented by a matrix $A$, column rank is the rank of the map and row rank is the rank of its dual. Since the dual map has kernel equal to the annihilator of the original image, rank-nullity gives equal ranks.

= Simultaneous basis exchange from a nonzero minor
{parent=Equality of row rank and column rank}

If the columns $v_1,\ldots,v_m$ of an $n$ by $m$ matrix are independent, some $m$ by $m$ row minor is nonzero. A nonzero term of that minor's determinant selects distinct nonzero coordinates $f(i)$, and replacing the corresponding standard basis vectors $e_{f(i)}$ by the $v_i$ leaves a basis.

= Duals of a subspace and its quotient
{parent=Dual map}

For a finite-dimensional vector space and $U\leq V$, dualizing the quotient and inclusion maps gives
$$
(V/U)^*\cong U^\circ,
\qquad
U^*\cong V^*/U^\circ.
$$

= Surjectivity of independent linear functionals
{parent=Dual space}

If $q_1,\ldots,q_n\in V^*$ are linearly independent, then
$$
x\longmapsto(q_1(x),\ldots,q_n(x))
$$
maps $V$ onto $F^n$. Otherwise a nonzero functional annihilating its proper image would give a nontrivial linear relation among the $q_j$.

= Polynomial moment functional
{parent=Dual space}

A polynomial moment functional evaluates a polynomial through a weighted integral, such as $p\mapsto\int p(t)w(t)dt$; finitely many moments determine it on a bounded-degree polynomial space.

= Rank of a matrix with affine columns
{parent=Linear algebra}

If the $j$th column of a matrix is $v+c_j\mathbf1$, then its image lies in $\operatorname{span}\{v,\mathbf1\}$. If $v$ is not constant and at least two $c_j$ differ, the rank is exactly two; for $m$ columns the nullity is $m-2$.

= Matrix determinant lemma
{parent=Linear algebra}
{wiki}

= Matrix inverse
{title2=$A^{-1}$}
{parent=Linear algebra}
{wiki}

A matrix inverse satisfies $AA^{-1}=A^{-1}A=I$.

= Matrix inverses
{synonym}

= Similar matrix
{parent=Linear algebra}
{wiki}

= Similarity transformation
{parent=Similar matrix}
{wiki=Matrix_similarity}

A similarity transformation sends a square matrix $A$ to $SAS^{-1}$ for an invertible matrix $S$. It changes the basis of a linear operator and preserves its characteristic polynomial and eigenvalues.

= Transpose similarity
{parent=Similar matrix}
{wiki}

Every square matrix over a field is similar to its transpose.

= Real similarity from complex similarity
{parent=Similar matrix}

If real matrices satisfy $C=SDS^{-1}$ for an invertible complex $S=X+iY$, then $CS=SD$, so both real matrices $X,Y$ intertwine $C$ and $D$. Since $\det(X+tY)$ is a nonzero real polynomial, some real $t$ makes $X+tY$ invertible and supplies a real similarity.

= Matrix trace
{parent=Linear algebra}
{wiki=Trace_(linear_algebra)}

The trace is the sum of diagonal entries,
$$
\operatorname{tr}A=\sum_iA_{ii}.
$$
Index interchange proves $\operatorname{tr}(AB)=\operatorname{tr}(BA)$ and hence cyclic invariance under any cyclic permutation of a product.

= Linear operator theory
{parent=Linear algebra}
{wiki=Operator_theory}

Linear operator theory studies <linear maps>, their <spectra>, invariant subspaces, and normal forms.

= Spectrum
{parent=Linear operator theory}
{wiki=Spectrum_(functional_analysis)}

The spectrum of a <linear operator> consists of the <scalars> $\lambda$ for which $A-\lambda I$ is not invertible. In finite dimensions it is the set of <eigenvalues>.

= Compression of a linear operator
{parent=Linear operator theory}

If $V$ is a subspace of a space with an <inner product> and $P_V$ is its <orthogonal projection onto a finite-dimensional subspace>[orthogonal projection], the compression of an operator $A$ to $V$ is
$$
P_VA|_V:V\longrightarrow V.
$$
If the columns of $Z$ are an <orthonormal basis> of $V$, its matrix in that basis is $Z^*AZ$.

= Self-adjoint operator
{parent=Linear operator theory}
{wiki}

An operator $T$ on an inner-product space is self-adjoint when
$$
\langle Tv,w\rangle=\langle v,Tw\rangle
$$
for all vectors $v,w$.

= Self-adjoint
{synonym}

= Finite-dimensional spectral theorem
{parent=Self-adjoint operator}
{wiki=Spectral_theorem}

Every self-adjoint operator on a finite-dimensional real inner-product space has an orthonormal basis of eigenvectors and only real eigenvalues.

= Orthonormal eigenbasis
{parent=Finite-dimensional spectral theorem}

An orthonormal eigenbasis of a <linear operator> is an <orthonormal basis> consisting of <eigenvectors>.

= Laguerre differential operator on polynomials
{parent=Self-adjoint operator}
{c}

On polynomials with weighted inner product
$$
\langle f,g\rangle=\int_0^\infty f(x)g(x)e^{-x}\,dx,
$$
the operator
$$
Lf=xf''+(1-x)f'
$$
is self-adjoint because $e^{-x}Lf=(xe^{-x}f')'$. Its eigenvalues on polynomials of degree at most $n$ are $0,-1,\ldots,-n$; its eigenvectors are scalar multiples of the <Laguerre polynomials>.

= Laguerre polynomial
{parent=Laguerre differential operator on polynomials}
{c}
{wiki}

= Generalized Laguerre polynomial
{parent=Laguerre polynomial}
{c}
{wiki}

The generalized Laguerre polynomial $L_n^{(\alpha)}$ is orthogonal on $[0,\infty)$ for the weight $x^\alpha e^{-x}$ when $\alpha>-1$.

= Rodrigues formula
{parent=Generalized Laguerre polynomial}
{c}
{wiki=Rodrigues%27_formula}

A Rodrigues formula constructs an orthogonal polynomial by differentiating a weight multiplied by a power. For generalized Laguerre polynomials,
$$
L_n^{(\alpha)}(x)
=\frac{x^{-\alpha}e^x}{n!}\frac{d^n}{dx^n}
\left(e^{-x}x^{n+\alpha}\right).
$$

= Spectra
{synonym}

= Real spectral theorem
{parent=Linear operator theory}
{wiki=Spectral_theorem}

Every real symmetric matrix is orthogonally diagonalizable. Distinct eigenspaces are orthogonal because symmetry gives $\lambda\langle u,v\rangle=\mu\langle u,v\rangle$.

= Power method
{parent=Linear operator theory}
{wiki=Power_iteration}

Iterate $q^{(k+1)}=Aq^{(k)}/\|Aq^{(k)}\|$. With a unique <dominant eigenvalue> separated by a <spectral gap> and a starting <vector> that is <nonorthogonal> to its <eigenvector>, the directions converge at the spectral-ratio rate.

= Quadratic Rayleigh-quotient improvement for the power method
{parent=Power method}

For a real symmetric matrix with $|\lambda_1|>|\lambda_2|$ and a starting vector having nonzero component in the leading eigendirection, power-method direction errors are
$$
O\!\left(\left|\frac{\lambda_2}{\lambda_1}\right|^k\right),
$$
whereas the <Rayleigh quotient> error is
$$
O\!\left(\left|\frac{\lambda_2}{\lambda_1}\right|^{2k}\right).
$$
The improvement occurs because orthogonality cancels terms linear in the direction error.

= Active spectrum of the power method
{parent=Power method}

The active spectrum consists of eigenvalues whose eigenspaces have nonzero projection of the starting vector. The power method converges toward the eigenvalue of largest modulus in this active spectrum; an exactly absent dominant component is never created in exact arithmetic.

= Inverse iteration
{parent=Power method}
{wiki=Inverse_iteration}

Inverse iteration repeatedly solves $(A-sI)y_{k+1}=q_k$ and normalizes. It is the power method for $(A-sI)^{-1}$ and converges toward an eigenvector whose eigenvalue is closest to the shift $s$.

= Convergence of fixed-shift inverse iteration
{parent=Inverse iteration}

If $A$ is real symmetric, $s$ is not an eigenvalue, one eigenvalue $\lambda_*$ is uniquely closest to $s$, and the initial vector has a nonzero component in its eigenspace, fixed-shift inverse iteration converges to that eigenspace. Its asymptotic direction-error ratio is
$$
\frac{|\lambda_*-s|}{|\lambda_{\rm next}-s|},
$$
where $\lambda_{\rm next}$ is second closest to $s$.

= Subspace iteration
{parent=Power method}
{wiki}

Subspace iteration repeatedly applies a matrix to several independent vectors and orthonormalizes them, converging to a dominant invariant subspace when a spectral gap separates the selected eigenvalue cluster.

= Dominant invariant subspace
{parent=Subspace iteration}

A dominant invariant subspace is spanned by eigenvectors whose eigenvalue moduli exceed those outside the subspace.

= Power method with a dominant eigenvalue cluster
{parent=Subspace iteration}

When several eigenvalues share the largest modulus, normalized power iterates need not converge individually, but every limit lies in their joint invariant subspace.

= Characteristic polynomial
{parent=Linear operator theory}
{wiki}

The characteristic polynomial is
$$
\chi_A(t)=\det(tI-A).
$$
Its roots are precisely the eigenvalues of $A$.

= Real parameter avoiding a singular matrix pencil
{parent=Characteristic polynomial}

For real matrices $A,B$, $p(t)=\det(A+tB)$ is a real polynomial. If $p(i)\ne0$, it is not the zero polynomial, so some real $\lambda$ satisfies $p(\lambda)\ne0$.

= Triangularization over an algebraically closed field
{parent=Characteristic polynomial}

Every square matrix over an algebraically closed field is similar to an upper-triangular matrix. Choose an eigenvector, extend it to a basis, and apply induction to the induced lower-right block.

= Eigenvalue interlacing
{parent=Linear operator theory}
{wiki}

Eigenvalue interlacing bounds the ordered eigenvalues of a compressed or rank-modified operator between those of the original operator.

= Jordan normal form
{parent=Linear operator theory}
{c}
{wiki}

= Algebraic multiplicity
{parent=Jordan normal form}
{wiki}

The algebraic multiplicity of an eigenvalue is its multiplicity as a root of the characteristic polynomial. In Jordan form it is the sum of the sizes of all blocks carrying that eigenvalue.

= Geometric multiplicity
{parent=Jordan normal form}
{wiki}

The geometric multiplicity of an eigenvalue is the dimension of its eigenspace. In Jordan form it equals the number of blocks carrying that eigenvalue.

= Jordan block
{parent=Jordan normal form}
{c}
{wiki}

A Jordan block has one eigenvalue on its diagonal and ones on its superdiagonal.

= Nilpotent Jordan block
{title2=$J_d(0)$}
{parent=Jordan block}
{c}

A nilpotent Jordan block has zeros on its diagonal and ones on its superdiagonal. Its $k$th power has ones on the $k$th superdiagonal, and a block of size $d$ satisfies $J_d(0)^d=0$.

= Jordan normal form of conjugation on two-by-two matrices
{parent=Jordan normal form}
{c}

If $\beta$ has eigenvalues $\lambda,\mu\ne0$ and is diagonalizable on a two-dimensional complex vector space, then
$$
\operatorname{JNF}(\phi_\beta)
=\operatorname{diag}\left(1,1,\frac\mu\lambda,\frac\lambda\mu\right).
$$
If $\beta$ has one size-two Jordan block, then
$$
\operatorname{JNF}(\phi_\beta)=J_3(1)\oplus J_1(1).
$$

= Characteristic and minimal polynomials determine similarity in dimension three
{parent=Jordan normal form}

Over an algebraically closed field, the characteristic polynomial gives the total size of the Jordan blocks for each eigenvalue, while the minimal polynomial gives the largest block size. For total dimension at most three, these two numbers determine the partition into block sizes: a multiplicity-three eigenvalue has partition $3$, $2+1$, or $1+1+1$ according as its largest block has size $3$, $2$, or $1$. Hence two matrices of size at most three with the same characteristic and minimal polynomials are similar.

= Generalized eigenvector
{parent=Jordan normal form}
{wiki}

A generalized eigenvector lies in the kernel of a positive power of A minus lambda I.

= Generalized eigenspaces for distinct eigenvalues form a direct sum
{parent=Generalized eigenvector}

For a linear operator $f$, the generalized eigenspaces
$$
\ker(f-\alpha I)^n
$$
belonging to distinct eigenvalues are linearly independent. On the generalized $\alpha$-eigenspace, $f-\beta I=(\alpha-\beta)I+N$ is invertible for $\beta\ne\alpha$ because the nilpotent part $N$ has a finite geometric-series inverse.

= Matrix exponential
{parent=Linear operator theory}
{wiki}

For a square matrix $A$, its exponential is the convergent power series
$$
e^{tA}=\sum_{k=0}^{\infty}\frac{t^kA^k}{k!}.
$$
It is the fundamental matrix for $y'=Ay$ and satisfies $e^{(s+t)A}=e^{sA}e^{tA}$.

= Laplace transform of a matrix exponential
{parent=Matrix exponential}

On its half-plane of convergence,
$$
\mathcal L\{e^{tA}\}(s)=(sI-A)^{-1}.
$$
Indeed, applying the transform to $y'=Ay$, $y(0)=y_0$, gives $(sI-A)Y=y_0$ for every $y_0$.

= Minimal polynomial
{parent=Linear operator theory}
{wiki}

The minimal polynomial $m_A$ is the unique monic polynomial of least degree satisfying $m_A(A)=0$. It divides every polynomial that annihilates $A$.

= Kernel decomposition for coprime polynomials
{parent=Minimal polynomial}

If $p_1,\ldots,p_r$ are pairwise coprime, then
$$
\ker\!\left(\prod_i p_i(A)\right)
=\bigoplus_i\ker p_i(A).
$$
Bezout identities between the factors construct the corresponding projections.

= Minimal polynomial bound from a triangular invariant flag
{parent=Minimal polynomial}

If an upper-triangular $n$ by $n$ matrix has diagonal entries $\lambda_1,\ldots,\lambda_n$, its standard invariant flag satisfies
$$
(A-\lambda_kI)V_k\subseteq V_{k-1}.
$$
The commuting product $\prod_k(A-\lambda_kI)$ therefore vanishes, proving $\deg m_A\leq n$ without invoking the full Cayley-Hamilton theorem.

= Jacobson lemma for a commuting commutator
{parent=Linear operator theory}
{c}

Let $C=[B,A]$ and suppose $AC=CA$. Then $C$ is nilpotent. For the derivation $D(X)=[B,X]$, one has $D(p(A))=p'(A)C$. If $f(A)=0$, induction gives
$$
f^{(k)}(A)C^{2^k-1}=0.
$$
Indeed, if $uC^m=C^mu=0$, differentiating $uC^m=0$ and multiplying on the left by $C^m$ gives $C^mD(u)C^m=0$; for $u=f^{(k)}(A)$ this replaces $m$ by $2m+1$. Taking $k=\deg f$ makes $f^{(k)}$ a nonzero constant and proves nilpotence.

= Translation finite-difference operator
{parent=Linear operator theory}

On all real-valued functions on $\mathbb R$, define $D_rf(x)=f(x+r)-f(x)$. The operators commute, and every real number except $-1$ is an eigenvalue. For eigenvalue $\lambda\ne-1$, values may be chosen independently on the cosets of $r\mathbb Z$ and extended by $f(x+r)=(1+\lambda)f(x)$, so every eigenspace is infinite-dimensional.

= Mixed finite difference of a polynomial
{parent=Translation finite-difference operator}

If $p$ has degree $n$ and leading coefficient $a_n$, then for nonzero steps $r_1,\ldots,r_n$,
$$
D_{r_1}\cdots D_{r_n}p=n!a_n\prod_{i=1}^nr_i.
$$
Consequently a degree-$n$ polynomial cannot be the sum of $n$ functions each periodic with some nonzero period.

= Nilpotent linear map
{parent=Linear operator theory}
{wiki}

A <linear map> $T:V\to V$ is nilpotent if $T^k=0$ for some <positive integer> $k$.

= Nilpotent operator
{synonym}

= Eigenvalue
{title2=$\lambda$}
{parent=Linear operator theory}
{wiki}

A <scalar> $\lambda$ is an eigenvalue of a <linear operator> $A$ when $Av=\lambda v$ for some nonzero <vector> $v$.

= Eigenvalues
{synonym}

= Eigenfunction
{parent=Eigenvalue}
{wiki}

An eigenfunction is a nonzero function $f$ satisfying $Lf=\lambda f$ for a linear operator $L$ and scalar <eigenvalue> $\lambda$.

= Eigenvector
{title2=$v$}
{parent=Eigenvalue}
{wiki}

A nonzero <vector> $v$ is an eigenvector of a <linear operator> $A$ when $Av=\lambda v$ for some <eigenvalue> $\lambda$.

= Eigenvectors
{synonym}

= Right eigenvector
{title2=$v$}
{parent=Eigenvector}

A right eigenvector satisfies $Av=\lambda v$.

= Left eigenvector
{title2=$u$}
{parent=Eigenvector}

A left eigenvector satisfies $u^TA=\lambda u^T$, equivalently $A^Tu=\lambda u$ over the real numbers.

= Eigenspace
{parent=Eigenvalue}
{wiki}

The eigenspace for $\lambda$ is $\ker(A-\lambda I)$.

= Eigenbasis
{parent=Eigenspace}
{wiki=Eigenbasis}

An eigenbasis is a <basis> consisting of <eigenvectors> of one operator.

= Simple eigenvalue
{parent=Eigenvalue}
{wiki=Simple_root}

An eigenvalue is simple when it has <algebraic multiplicity> one.

= Eigenvalue sensitivity
{title2=$s(\lambda)$}
{parent=Simple eigenvalue}

For unit <left eigenvector> $u$ and <right eigenvector> $v$ of a real matrix at a <simple eigenvalue> $\lambda$, its normwise sensitivity is
$$
s(\lambda)=\frac1{|u^Tv|}.
$$
It measures the first-order amplification of a matrix perturbation into an eigenvalue perturbation.

= First-order perturbation of a simple eigenvalue
{parent=Eigenvalue sensitivity}

If $A(t)=A+tE$ and $\lambda(t)$ is a <differentiable function> defining a simple eigenvalue, then
$$
\lambda'(0)=\frac{u^TEv}{u^Tv}.
$$
The <Cauchy-Schwarz inequality> and the <operator norm> give
$$
|\lambda'(0)|\leq\|E\|_2s(\lambda).
$$

= Diagonalizable matrix
{parent=Eigenvalue}
{wiki=Diagonalizable_matrix}

A matrix is diagonalizable exactly when its eigenvectors span the whole vector space.

= Diagonalizability inherited from an invertible power
{parent=Diagonalizable matrix}

Let $A$ be invertible over an algebraically closed field. If $A^m$ is diagonalizable for some $m>0$, then $A$ is diagonalizable. Indeed, a nontrivial Jordan block $A|_E=\lambda I+N$ has $\lambda\ne0$ and
$$
(\lambda I+N)^m=\lambda^mI+m\lambda^{m-1}N+\cdots.
$$
The first nonzero nilpotent term prevents this power from being diagonalizable.

= Complex similarity of real matrices implies real similarity
{parent=Diagonalizable matrix}

Suppose real matrices $A,B$ satisfy $AP=PB$ for an invertible complex matrix $P=X+iY$. Then $AX=XB$ and $AY=YB$. The real polynomial $\det(X+tY)$ is not identically zero because its value at $t=i$ is nonzero, so some real $t$ makes $X+tY$ invertible. This real matrix intertwines $A$ and $B$.

= Distinct eigenvalues imply diagonalizability
{parent=Diagonalizable matrix}

Eigenvectors belonging to distinct eigenvalues are linearly independent. Consequently, an $n$-dimensional operator with $n$ distinct eigenvalues is diagonalizable.

= Spectral decomposition
{parent=Diagonalizable matrix}
{wiki}

A spectral decomposition expresses a <linear operator> through its <eigenvalues> and projections onto its <eigenspaces>.

= Dominant eigenvalue
{parent=Spectral decomposition}
{wiki}

An <eigenvalue> is dominant when its <modulus> is greater than or equal to that of every other eigenvalue. It is uniquely dominant when the inequality is strict.

= Spectral gap
{parent=Spectral decomposition}
{wiki}

A spectral gap is a positive separation between designated parts of the <spectrum> of a <linear operator>. For the <power method>, the relevant gap separates the modulus of the <dominant eigenvalue> from the other eigenvalue moduli.

= Adjoint matrix
{parent=Linear operator theory}
{wiki=Conjugate_transpose}

The adjoint of a complex matrix is its conjugate transpose,
$$
A^\dagger=\overline A^T.
$$

= Unitary matrix
{parent=Linear operator theory}
{wiki}

A complex matrix is unitary when $U^\dagger U=UU^\dagger=I$. Its columns form an orthonormal basis, and it preserves inner products and norms.

= Normal matrix
{parent=Linear operator theory}
{wiki}

A matrix is normal when $AA^\dagger=A^\dagger A$.

= Unitary diagonalization of a normal matrix
{parent=Normal matrix}
{wiki=Spectral_theorem\#Normal_matrices}

Every finite-dimensional complex normal matrix has an orthonormal eigenbasis, equivalently
$$
U^\dagger AU=D
$$
for some unitary $U$ and diagonal $D$.

= Schur triangularization
{parent=Linear operator theory}
{c}
{wiki=Schur_decomposition}

Every complex square matrix is unitarily similar to an upper triangular matrix. The proof selects an eigenvector, extends it to an orthonormal basis, and applies induction to the compression on its orthogonal complement.

= Unilateral shift operator
{parent=Linear operator theory}
{wiki=Shift_operator}

The unilateral shifts on sequences move every component one place left or right. Their spectra illustrate behavior absent in finite dimensions.

= Point spectrum
{parent=Unilateral shift operator}
{wiki=Spectrum_(functional_analysis)\#Point_spectrum}

The point spectrum of an operator is its set of eigenvalues.

= Wandering-vector characterization of a unilateral shift
{parent=Unilateral shift operator}

An operator $T$ is a unilateral shift exactly when it is an isometry, $(\operatorname{Im}T)^\perp$ is one-dimensional, and $\bigcap_{n\geq1}\operatorname{Im}(T^n)=\{0\}$. A unit vector $e_1$ in the first orthogonal complement is wandering, and $e_n=T^{n-1}e_1$ is a complete orthonormal basis.

= Rayleigh quotient
{title2=$R[\psi]$}
{parent=Linear operator theory}
{c}
{wiki}

For a self-adjoint operator $H$, the Rayleigh quotient is
$$
R[\psi]=\frac{\langle\psi|H|\psi\rangle}{\langle\psi|\psi\rangle}.
$$
For a symmetric matrix this reduces to $x^TAx/(x^Tx)$.

= Rayleigh quotient iteration
{parent=Rayleigh quotient}
{c}
{wiki}

Starting from a unit vector $v_k$, Rayleigh quotient iteration sets $\mu_k=R[v_k]$, solves
$$
(A-\mu_kI)w_k=v_k,
$$
and normalizes $v_{k+1}=w_k/\|w_k\|_2$. The shift $\mu_k$ is updated from the new vector.

= Rayleigh-Ritz variational principle
{parent=Rayleigh quotient}
{c}
{wiki=Rayleigh_quotient\#Bounds_for_Hermitian_matrices}

The extremal Rayleigh quotients of a self-adjoint operator are the extremal spectral values. In particular, a Hamiltonian with a discrete spectrum satisfies $R[\psi]\geq E_0$, with equality precisely on its ground-state eigenspace. More generally, minimizing over states orthogonal to the first $j$ eigenspaces gives an upper bound on $E_j$.

= Odd-state variational principle for an even potential
{parent=Rayleigh-Ritz variational principle}

For a one-dimensional even potential, the ground state is even and the first excited state is odd. Every normalized odd trial wavefunction is therefore orthogonal to the ground state and gives
$$
E_1\leq\langle\psi_{\rm odd},H\psi_{\rm odd}\rangle.
$$

= Gaussian variational bound for an attractive Gaussian well
{c}
{parent=Rayleigh-Ritz variational principle}

For
$$
H=-\frac{d^2}{dx^2}-V_0e^{-x^2}
$$
and the trial state $\psi_a(x)=e^{-ax^2/2}$,
$$
E_0\leq E(a)
=\frac a2-V_0\sqrt{\frac a{1+a}}.
$$
This is negative for some $a>0$ for every $V_0>0$, proving that the attractive well has a bound state. For small $V_0$, taking $a=V_0^2$ gives
$$
-V_0<E_0\leq-\frac12V_0^2+O(V_0^4).
$$

= Finite-subspace variational method
{parent=Rayleigh-Ritz variational principle}

For an orthonormal trial set $\phi_1,\ldots,\phi_N$, the Rayleigh quotient on their span is
$$
\frac{\alpha^\dagger\mathcal H\alpha}{\alpha^\dagger\alpha},
\qquad
\mathcal H_{nm}=\langle\phi_n|H|\phi_m\rangle.
$$
Its minimum is the smallest eigenvalue of $\mathcal H$, which is therefore the optimal variational upper bound available in that trial subspace.

= Two-mode variational bound for a linearly tilted square well
{parent=Finite-subspace variational method}

For the first two sine modes in a width-$a$ infinite well with
$V(x)=9\hbar^2x/(ma^3)$, the Hamiltonian in units $\hbar^2/(ma^2)$ is
$$
\begin{pmatrix}
(\pi^2+9)/2&-16/\pi^2\\
-16/\pi^2&(4\pi^2+9)/2
\end{pmatrix}.
$$
Its lower eigenvalue gives the upper bound
$$
E_0\leq\frac{\hbar^2}{ma^2}
\left[
\frac{5\pi^2+18}{4}
-\sqrt{\frac{9\pi^4}{16}+\frac{256}{\pi^4}}
\right].
$$

= Cyclic vector
{parent=Linear operator theory}
{wiki}

A cyclic vector has iterates under an operator that span the whole vector space.

= Companion matrix
{parent=Linear operator theory}
{wiki}

A companion matrix represents multiplication by x in the power basis modulo a monic polynomial.

= Invariant direct-sum decomposition
{parent=Linear operator theory}
{wiki}

An invariant direct-sum decomposition splits a space into nonzero subspaces preserved by the operator.

= Rational canonical form
{parent=Linear operator theory}
{wiki}

Rational canonical form expresses a linear map as a direct sum of companion matrices of invariant factors, and works over any field.

= Multilinear algebra
{parent=Linear algebra}
{wiki}

Multilinear algebra studies maps linear in each of several arguments, including determinants and tensor products.

= Tensor
{parent=Multilinear algebra}
{wiki}

A rank-$n$ tensor on a vector space $V$ is a multilinear map of $n$ vector or covector arguments, equivalently an element of an $n$-fold tensor product. Its components transform by one copy of the change-of-basis matrix for each index.

= Covariant tensor
{parent=Tensor}
{wiki=Covariance_and_contravariance_of_vectors}

A covariant tensor is multilinear in vector arguments and transforms with one covariant index for each argument.

= Antisymmetric second-rank tensor
{parent=Tensor}
{wiki=Skew-symmetric_matrix}

A second-rank Cartesian tensor is antisymmetric when $T_{ij}=-T_{ji}$. Orthogonal coordinate changes preserve this property because $T'=RTR^T$ implies $(T')^T=RT^TR^T=-T'$.

= Totally antisymmetric tensor
{parent=Tensor}
{wiki=Antisymmetric_tensor}

A tensor changes sign whenever two indices are interchanged. In three dimensions, a rank-two antisymmetric tensor is $A_{ij}=a_k\epsilon_{kij}$, a rank-three one is $A_{ijk}=a\epsilon_{ijk}$, and every totally antisymmetric tensor of rank greater than three vanishes.

= Isotropic tensor
{parent=Tensor}
{wiki}

An isotropic tensor is invariant under every proper orthogonal change of basis. A rank-four isotropic tensor has the form
$$
T_{ijkl}=a\delta_{ij}\delta_{kl}
+b\delta_{ik}\delta_{jl}
+c\delta_{il}\delta_{jk}.
$$
Rank-five isotropic pseudotensors are formed by multiplying one Kronecker delta by one Levi-Civita symbol and permuting the five indices.

= Einstein notation
{parent=Tensor}
{c}
{wiki}

Einstein notation sums automatically over an index repeated once up and once down, or twice in Euclidean coordinates.

= Einstein summation convention
{c}
{synonym}

= Tensor contraction
{parent=Multilinear algebra}
{wiki}

= Reciprocal basis
{parent=Multilinear algebra}
{wiki=Reciprocal_lattice\#Reciprocal_basis}

For a basis $e_1,e_2,e_3$ with scalar triple product $\Delta$, the reciprocal vectors are cyclic cross products divided by $\Delta$ and satisfy $f_i\cdot e_j=\delta_{ij}$.

= Scalar triple product
{parent=Multilinear algebra}
{wiki}

The scalar triple product $a\cdot(b\times c)$ is the signed volume of the parallelepiped spanned by three vectors.

= Oriented volume
{parent=Scalar triple product}
{wiki=Orientation_(vector_space)}

An oriented volume retains the sign of a determinant; reversing the orientation changes that sign.

= Determinant
{title2=$\det$}
{parent=Multilinear algebra}
{wiki}

The determinant is the alternating multilinear volume scale of a square matrix; it is nonzero exactly for invertible matrices.

= Determinants
{synonym}

= Multilinearity of the determinant
{parent=Determinant}

The determinant is linear in each row and in each column when all the others are held fixed.

= Leibniz formula for determinants
{parent=Determinant}
{c}
{wiki=Leibniz_formula_for_determinants}

For an $n$ by $n$ matrix,
$$
\det A=\sum_{\sigma\in S_n}\operatorname{sgn}(\sigma)
\prod_{i=1}^na_{i,\sigma(i)}.
$$

= Tridiagonal determinant recurrence
{parent=Determinant}

For diagonal entries $a_i$, superdiagonal entries $b_i$, and subdiagonal entries $c_i$, the leading principal determinants satisfy
$$
d_n=a_nd_{n-1}-b_{n-1}c_{n-1}d_{n-2},
\qquad d_0=1,\quad d_1=a_1.
$$

= Linear system
{parent=Linear algebra}
{wiki=System_of_linear_equations}

A linear system $Ax=b$ is consistent when $b$ lies in the column space of $A$; its solutions form a translate of $\ker A$.

= Linear equation
{parent=Linear system}
{wiki}

A linear equation is an equation in which the unknowns occur only to the first power and are combined linearly.

= Fredholm alternative for a matrix
{parent=Linear system}
{c}

The system $Ax=b$ is solvable exactly when $b$ is orthogonal to every vector in $\ker A^T$.

= Symmetric and antisymmetric parts of a matrix
{parent=Linear algebra}

Every square matrix decomposes uniquely as $A=(A+A^T)/2+(A-A^T)/2$, a symmetric part plus an antisymmetric part.

= Symmetric matrix
{parent=Linear algebra}
{wiki}

A real matrix is symmetric when $A^T=A$. It represents a <self-adjoint operator> in the standard Euclidean <inner product>.

= Axis-angle decomposition
{parent=Linear algebra}
{wiki=Axis–angle_representation}

An axis-angle map fixes or scales one axis and acts on its perpendicular plane by a rotation, possibly combined with a dilation.

= Inverse of an axis-angle linear map
{parent=Axis-angle decomposition}

Invert the axial scalar and invert the perpendicular complex scalar $\alpha+i\gamma$ as $(\alpha-i\gamma)/(\alpha^2+\gamma^2)$.

= Reflection matrix
{parent=Linear algebra}
{wiki=Reflection_(mathematics)}

A plane reflection with unit normal $n$ is $I-2nn^T$; it fixes the plane $n^\perp$ and negates $n$.

= Reflection in a hyperplane
{parent=Reflection matrix}

For a nonzero normal vector $a$, reflection in $a\cdot x=0$ has matrix
$$
I-\frac{2aa^T}{a^Ta}.
$$
Its eigenvalues are $-1$ along $a$ and $1$ on $a^\perp$, so its determinant is $-1$.

= Composition of two plane reflections
{parent=Reflection matrix}
{wiki=Reflection_(mathematics)#Composition_of_reflections}

The composition of reflections in two planes through the origin is a rotation about their line of intersection through twice the oriented angle between the planes.

= Rotation matrix
{parent=Linear algebra}
{wiki}

A rotation matrix is an orthogonal matrix with determinant $1$.

= Rotational symmetry
{parent=Rotation matrix}
{wiki}

An object has rotational symmetry when a nontrivial <rotation matrix> leaves it unchanged.

= Cartesian coordinate system
{parent=Linear algebra}
{c}
{wiki}

A Cartesian coordinate system specifies a point by its signed components along mutually perpendicular coordinate axes.

= Positive quadrant
{parent=Cartesian coordinate system}
{wiki=Quadrant_(plane_geometry)}

The positive quadrant of the Cartesian plane is the set of points $(x,y)$ with $x>0$ and $y>0$.

= Matrix geometric series
{parent=Linear algebra}

For invertible $I-A$, $\sum_{j=0}^{n-1}A^j=(I-A^n)(I-A)^{-1}$.

= Adjugate matrix
{parent=Linear algebra}
{wiki}

The adjugate is the transpose of the cofactor matrix and satisfies $A\operatorname{adj}(A)=(\det A)I$.

= Matrix equivalence
{parent=Linear algebra}
{wiki}

Equivalent matrices represent the same linear map after independent changes of domain and codomain bases.

= Rank inequality for a composition
{parent=Linear algebra}
{wiki}

For maps through a finite-dimensional intermediate space V, rank(alpha beta) is at least rank(alpha)+rank(beta)-dim(V).

= Schur complement
{parent=Linear algebra}
{c}
{wiki}

For an invertible block P, the Schur complement of P in a block matrix is $S-RP^{-1}Q$.

= Orthogonal similarity
{parent=Linear algebra}
{wiki}

Orthogonal similarity maps A to QAQ^T and preserves eigenvalues and symmetry.

= Product of symmetric matrices
{parent=Linear algebra}
{wiki}

Every complex square matrix is a product of two symmetric matrices, one invertible.

= Inner product
{title2=$\langle\cdot,\cdot\rangle$}
{parent=Linear algebra}
{wiki}

= Orthonormal basis
{parent=Inner product}
{wiki}

An orthonormal basis is a <basis> whose vectors are pairwise <orthogonal vectors>[orthogonal] and each have <normed vector space>[norm] one.

= Gram matrix
{parent=Inner product}
{c}
{wiki}

For vectors $v_1,\ldots,v_n$, the Gram matrix is
$$
G_{ij}=\langle v_i,v_j\rangle.
$$
Two finite vector families with the same Gram matrix are related by an isometry between their spans; in finite dimensions that isometry can be extended to a unitary map after completing orthonormal bases.

= Pythagorean theorem in an inner-product space
{parent=Inner product}

If $u$ and $v$ are orthogonal, then
$$
\lVert u+v\rVert^2=\lVert u\rVert^2+\lVert v\rVert^2.
$$

= Orthogonal projection onto a finite-dimensional subspace
{parent=Inner product}

If $e_0,\ldots,e_n$ is an orthogonal basis of a finite-dimensional subspace $W$, the unique closest point in $W$ to $f$ is
$$
P_Wf=\sum_{k=0}^n\frac{\langle f,e_k\rangle}{\langle e_k,e_k\rangle}e_k.
$$
The residual $f-P_Wf$ is orthogonal to $W$.

= Orthogonal projection matrix
{parent=Orthogonal projection onto a finite-dimensional subspace}
{wiki=Projection_(linear_algebra)}

A complex matrix $P$ is an orthogonal projection exactly when
$$
P^2=P=P^\dagger.
$$
Its image and kernel are orthogonal complements.

= Least-squares polynomial in an orthogonal-polynomial basis
{parent=Orthogonal projection onto a finite-dimensional subspace}

For orthogonal polynomials $Q_0,\ldots,Q_n$ under a positive weighted integral inner product, the least-squares approximation to $f$ of degree at most $n$ is
$$
p_n^*=\sum_{k=0}^n\frac{\langle f,Q_k\rangle}{\langle Q_k,Q_k\rangle}Q_k.
$$
Its residual is orthogonal to every polynomial of degree at most $n$.

= Orthogonal vectors
{parent=Inner product}
{wiki=Orthogonality}

Two <vectors> are orthogonal when their <inner product> is zero.

= Orthogonal
{synonym}

= Nonorthogonal vectors
{parent=Inner product}

Two <vectors> are nonorthogonal when their <inner product> is nonzero. In the <power method>, the starting vector must be nonorthogonal to a dominant <eigenvector>.

= Nonorthogonal
{synonym}

= Gram-Schmidt process
{parent=Inner product}
{c}
{wiki}

The Gram-Schmidt process replaces linearly independent vectors $v_1,\ldots,v_n$ by orthogonal vectors with the same successive spans, subtracting from each $v_k$ its projections onto the preceding vectors.

= QR decomposition
{title2=$A=QR$}
{c}
{parent=Gram-Schmidt process}
{wiki=QR_decomposition}

A reduced QR decomposition of a full-column-rank matrix writes $A=QR$, where $Q$ has orthonormal columns and $R$ is upper triangular with positive diagonal.

= Gram-Schmidt orthogonalization for a symmetric bilinear form
{parent=Gram-Schmidt process}

For a symmetric bilinear form $B$, the same projection formula
$$
u_k=v_k-\sum_{j<k}\frac{B(v_k,u_j)}{B(u_j,u_j)}u_j
$$
produces a $B$-orthogonal basis whenever the chosen pivots $B(u_j,u_j)$ are nonzero. Reordering the input vectors can avoid a zero pivot when a suitable anisotropic vector remains.

= Quadratic form
{parent=Linear algebra}
{wiki}

A quadratic form is a homogeneous degree-two function $Q(v)$ represented in coordinates by $v^TAv$ with $A$ symmetric.

= Rank of a quadratic form
{parent=Quadratic form}

The rank of a quadratic form is the rank of its associated symmetric bilinear form.

= Rank
{synonym}

= Signature of a quadratic form
{title2=$\sigma$}
{parent=Quadratic form}
{wiki=Signature_(linear_algebra)}

The signature of a real quadratic form is the number of positive squares minus the number of negative squares in its diagonal normal form.

= Signature
{synonym}

= Polarization identity
{parent=Quadratic form}
{wiki}

Over a field of characteristic other than two, a quadratic form recovers its symmetric bilinear form by
$$B(u,v)=\frac12\bigl(Q(u+v)-Q(u)-Q(v)\bigr).$$

= Polarization argument for a vanishing quadratic form
{parent=Polarization identity}

For a complex sesquilinear form $B$, the values $B(x,x)$ determine $B(x,y)$. In particular, expanding at $x+y$ and $x+iy$ shows that $B(x,x)=0$ for every $x$ implies $B=0$. Over the reals this conclusion can fail for nonsymmetric bilinear forms, as a nonzero skew-symmetric form has zero diagonal.

= Positive semidefinite bilinear form
{parent=Quadratic form}
{wiki=Definite_matrix}

A symmetric bilinear form is positive semidefinite when $B(v,v)\geq0$ for every $v$, and positive definite when equality is possible only for $v=0$.

= Positive semidefinite matrix
{parent=Quadratic form}
{wiki=Definite_matrix}

A real symmetric or complex Hermitian matrix $A$ is positive semidefinite when $x^\dagger Ax\geq0$ for every <vector> $x$, equivalently when all its <eigenvalue>[eigenvalues] are nonnegative.

= Principal square root of a positive semidefinite matrix
{title2=$\sqrt A$}
{parent=Positive semidefinite matrix}
{wiki=Square_root_of_a_matrix#Positive_semidefinite_matrices}

If $A=Q\operatorname{diag}(\lambda_1,\ldots,\lambda_n)Q^T$ is real symmetric and positive semidefinite, its principal square root is
$$
\sqrt A=Q\operatorname{diag}(\sqrt{\lambda_1},\ldots,\sqrt{\lambda_n})Q^T.
$$
It is symmetric and positive semidefinite, and is positive definite exactly when $A$ is.

= Polar decomposition of an invertible real matrix
{parent=Principal square root of a positive semidefinite matrix}
{wiki=Polar_decomposition}

Every invertible real matrix has the polar decomposition
$$
M=RP,
\qquad
P=\sqrt{M^TM},
\qquad
R=MP^{-1},
$$
where $P$ is symmetric positive definite and $R$ is orthogonal.

= Positive-definite matrix
{parent=Quadratic form}
{wiki=Definite_matrix}

A real symmetric matrix $A$ is positive definite when
$$
x^TAx>0
$$
for every nonzero real <vector> $x$. It therefore defines the <inner product> $\langle x,y\rangle_A=x^TAy$ and is invertible.

= Sylvester's criterion
{c}
{parent=Positive-definite matrix}
{wiki}

A Hermitian matrix is positive definite exactly when all its leading principal minors are positive.

= Spectral condition number of a positive-definite matrix
{title2=$\kappa_2(A)$}
{parent=Positive-definite matrix}
{wiki=Condition_number}

For a real symmetric positive-definite matrix,
$$
\kappa_2(A)=\frac{\lambda_{\max}(A)}{\lambda_{\min}(A)}.
$$
It measures the sensitivity of the linear system and controls convergence bounds for iterative solvers.

= Condition number
{synonym}

= Hermitian positive-definite matrix
{parent=Positive-definite matrix}
{wiki=Definite_matrix}

A complex matrix $A$ is Hermitian positive definite when $A^*=A$ and $z^*Az>0$ for every nonzero complex vector $z$.

= Toeplitz matrix
{parent=Hermitian positive-definite matrix}
{c}
{wiki}

A Toeplitz matrix is constant along each diagonal, so its entries have the form $T_{nm}=t_{n-m}$.

= Sylvester's law of inertia
{parent=Quadratic form}
{c}
{wiki=Sylvester%27s_law_of_inertia}

Every real quadratic form is congruent to a diagonal form with entries $+1,-1,0$, and the numbers of entries of each kind are invariant.

= Totally isotropic subspace
{parent=Quadratic form}
{wiki=Isotropic_quadratic_form}

A subspace $E$ is totally isotropic for $B$ when $B(u,v)=0$ for all $u,v\in E$. For a real <nondegenerate bilinear form> of inertia $(p,q)$, its dimension is at most $\min(p,q)$.

= Isotropic vector
{parent=Quadratic form}
{wiki=Isotropic_quadratic_form}

A vector $v$ is isotropic for a quadratic form $q$ when $q(v)=0$.

= Quadratic form gradient
{parent=Quadratic form}

For symmetric $A$, the quadratic form $x^TAx$ has gradient $2Ax$.

= Schur product theorem
{parent=Linear algebra}
{c}
{wiki}

The entrywise product of two positive semidefinite matrices is positive semidefinite. Writing each matrix as a sum of rank-one Gram matrices reduces the claim to $(uu^T)\circ(vv^T)=(u\circ v)(u\circ v)^T$.

= Orthogonal group
{title2=$O(n)$}
{parent=Linear algebra}
{wiki}

The real orthogonal group is $O(n)=\{R:R^TR=I\}$. Its tangent space at the identity is the vector space of skew-symmetric matrices.

= Orthogonal matrix
{parent=Orthogonal group}
{wiki}

A real square matrix is orthogonal when $R^TR=I$, equivalently when its columns form an orthonormal basis.

= Special orthogonal group
{parent=Orthogonal group}
{wiki}

The special orthogonal group is
$$
SO(n)=\{A\in M_n(\mathbb R):A^TA=I,\ \det A=1\}.
$$
Every element of $SO(2)$ is a rotation in the plane. Every element of $SO(3)$ fixes an axis and restricts to a planar rotation on its perpendicular plane.

= Skew-symmetric matrix
{parent=Orthogonal group}
{wiki}

A matrix is skew-symmetric when $A^T=-A$.

= Cross-product matrix
{parent=Skew-symmetric matrix}

For $a=(a_1,a_2,a_3)^T$, the map $b\mapsto a\times b$ has matrix
$$
[a]_\times=
\begin{pmatrix}
0&-a_3&a_2\\
a_3&0&-a_1\\
-a_2&a_1&0
\end{pmatrix}.
$$
It has trace zero and annihilates $a$, so its determinant is zero.

= Tensor product of linear maps
{parent=Linear algebra}
{wiki=Tensor_product\#Tensor_product_of_linear_maps}

The tensor product is determined by $(A\otimes B)(x\otimes y)=Ax\otimes By$ and linear extension.

= Trace-duality bound
{parent=Linear algebra}

For $M\succeq0$ with $\operatorname{tr}M\leq s$, $\operatorname{tr}(MA)\leq s\max(\lambda_{\max}(A),0)\leq s\lVert A\rVert_{\rm op}$.

= Diagonal matrix
{parent=Linear algebra}
{wiki}

A diagonal matrix has zero entries away from its main diagonal. Its eigenvalues are its diagonal entries.

= Block-diagonal matrix
{parent=Diagonal matrix}
{wiki=Block_matrix}

A block-diagonal matrix has square matrices along its block diagonal and zero off-diagonal blocks. Its determinant is the product of the block determinants.

= Triangular matrix
{parent=Linear algebra}
{wiki}

An upper or lower triangular matrix has all entries on one side of the main diagonal equal to zero. Its eigenvalues, with algebraic multiplicity, are its diagonal entries.

= Square root
{title2=$\sqrt{x}$}
{parent=Algebra}
{wiki}

A square root of an element $x$ is an element $y$ such that $y^2=x$. The notation $\sqrt{x}$ denotes the nonnegative square root when $x$ is a <real number>[nonnegative real number].

= Algebraic number
{parent=Algebra}
{wiki}

= Countability of algebraic numbers
{parent=Algebraic number}

Integer polynomials form a countable set and each has finitely many roots, so the algebraic numbers are countable.

= Bezout identity
{parent=Algebra}
{c}
{wiki}

= Countable set
{parent=Algebra}
{wiki}

= Countable union of countable sets
{parent=Countable set}
{wiki}

A countable union of countable sets is countable, using a diagonal enumeration.

= Cantor pairing function
{parent=Countable set}
{c}
{wiki=Pairing_function}

The Cantor pairing function is an explicit bijection between $\mathbb N^2$ and $\mathbb N$, obtained by enumerating lattice points along successive diagonals.

= Cantor's diagonal argument
{parent=Countable set}
{c}
{wiki=Cantor%27s_diagonal_argument}

Cantor's diagonal argument proves that the set of infinite binary sequences is uncountable: a sequence obtained by changing the $n$th digit of the $n$th listed sequence differs from every sequence in the list.

= Eigenfunction expansion
{parent=Algebra}
{wiki}

= Field extension
{parent=Algebra}
{wiki}

= Tower law
{parent=Field extension}
{wiki=Degree_of_a_field_extension#The_multiplicativity_formula_for_degrees}

For finite extensions $K\subseteq L\subseteq M$,
$$
[M:K]=[M:L][L:K].
$$

= Field compositum
{parent=Field extension}
{wiki=Compositum_(field_theory)}

Inside a common overfield, the compositum $L_1L_2$ is the smallest field containing both $L_1$ and $L_2$.

= Linearly disjoint field extensions
{parent=Field compositum}
{wiki=Linearly_disjoint}

Finite extensions $L_1/K$ and $L_2/K$ are linearly disjoint when products of basis elements form a basis of their compositum. In particular,
$$
[L_1L_2:K]=[L_1:K][L_2:K].
$$
Extensions of coprime finite degrees are linearly disjoint.

= Algebraic closure
{parent=Field extension}
{wiki}

An algebraic closure of a field $K$ is an algebraic extension $\overline K/K$ for which $\overline K$ is algebraically closed.

= Intermediate field
{parent=Field extension}
{wiki}

For a field extension $L/K$, an intermediate field is a field $M$ satisfying $K\subseteq M\subseteq L$.

= Quadratic extension
{parent=Field extension}
{wiki}

A quadratic extension is a field extension of degree two. It is normal because the other root of the minimal polynomial $T^2+bT+c$ of any non-base-field element is $-b-\alpha$ and lies in the same extension.

= Transitivity of algebraic extensions
{parent=Field extension}
{wiki=Algebraic_extension}

If $L/K$ and $K/F$ are algebraic field extensions, then $L/F$ is algebraic. In particular, an element algebraic over an algebraic extension of $F$ is algebraic over $F$.

= Finite field
{title2=$\mathbb F_q$}
{parent=Algebra}
{wiki}

A finite field is a field with finitely many elements; its size is a prime power.

= Multiplicative group of a finite field is cyclic
{parent=Finite field}

The nonzero elements of every finite field form a cyclic group. For a field with $q$ elements, combine elements realizing each prime-power factor of the group exponent to obtain an element of exponent order; the root bound for $X^m-1$ then forces that order to be $q-1$.

= Finite multiplicative subgroup of a field is cyclic
{parent=Multiplicative group of a finite field is cyclic}

Every finite <subgroup> $H$ of the multiplicative group of a <field> is <cyclic group>[cyclic]. If $m$ is the exponent of $H$, commutativity lets one combine elements whose orders realize the prime-power factors of $m$, producing an element of order $m$. Every element of $H$ is a root of $X^m-1$, so the <Lagrange root bound over a field> gives $|H|\leq m$. Since $m\leq|H|$, equality holds and that element generates $H$.

= Power roots in a finite field
{parent=Multiplicative group of a finite field is cyclic}

In a finite field with $q$ elements, the equation $x^k=1$ has exactly $\gcd(k,q-1)$ solutions. Indeed, after choosing a generator $g$ of the multiplicative group, $x=g^j$ is a solution exactly when $(q-1)\mid kj$.

= Index-two square-class argument for three related residues
{parent=Multiplicative group of a finite field is cyclic}

For odd $p$, the nonzero squares form an index-two subgroup of $\mathbb F_p^\times$. Given nonzero $a,b$, either $a$ or $b$ is a square, or both occupy the nontrivial coset and $ab$ is a square. Taking $a=2$ and $b=3$ shows that one of $2,3,6$ is a square, apart from the immediate zero case $p=3$.

= Function composition
{title2=$\circ$}
{parent=Algebra}
{wiki}

= Injective function
{parent=Algebra}
{wiki}

= Irrational number
{parent=Algebra}
{wiki}

= Irrotational vector field
{parent=Algebra}
{wiki}

= Linear recurrence relation
{parent=Algebra}
{wiki}

= Ordinary generating function of a recurrence
{parent=Linear recurrence relation}

For a sequence $(u_n)$, the ordinary generating function is
$$
G(x)=\sum_{n\geq0}u_nx^n.
$$
Multiplying a linear recurrence by $x^n$ and summing converts shifts of the sequence into polynomial multiples of $G$.

= Floor-half generating function
{parent=Ordinary generating function of a recurrence}

The sequence $\lfloor n/2\rfloor$ has generating function
$$
\sum_{n\geq1}\left\lfloor\frac n2\right\rfloor x^n
=\frac{x^2}{(1-x)^2(1+x)}.
$$

= Affine recurrence
{parent=Linear recurrence relation}

An affine recurrence has the form $x_{n+1}=Ax_n+b$ and is solved by a matrix geometric sum.

= Stable affine recurrence
{parent=Affine recurrence}

If the spectral radius of $A$ is below one, $x_{n+1}=Ax_n+b$ converges to the unique fixed point $(I-A)^{-1}b$.

= Linearization
{parent=Algebra}
{wiki}

Linearization replaces a nonlinear relation near a reference state by its first-order approximation.

= Linearization of a dynamical system
{parent=Linearization}
{wiki}

= Mathematical induction
{parent=Algebra}
{wiki}

= Strong induction
{parent=Mathematical induction}
{wiki}

Strong induction allows the step for an integer to use the result for every smaller integer.

= Backward induction
{parent=Mathematical induction}
{wiki}

Backward induction proves or computes a finite sequence of statements by starting at its terminal index and stepping toward the initial index.

= Minimal number of generators
{parent=Algebra}
{wiki}

= Nilpotent polynomial
{parent=Algebra}
{wiki}

= Noetherian ring
{parent=Algebra}
{c}
{wiki}

= Ascending chain condition
{parent=Noetherian ring}
{wiki=Ascending_chain_condition}

The ascending chain condition requires every chain
$$
I_1\subseteq I_2\subseteq\cdots
$$
of ideals to stabilize. A ring is Noetherian exactly when it satisfies this condition, equivalently when every ideal is finitely generated.

= Noetherianity from finite additive rank
{parent=Noetherian ring}

If a ring $R$ is finitely generated as an abelian group, then it is Noetherian. Every ideal is an additive subgroup of that finitely generated abelian group and is therefore generated by finitely many elements as an abelian group. The same elements generate it as an $R$-ideal because integer multiples are already scalar multiples by elements of $R$.

= Hilbert basis theorem
{parent=Noetherian ring}
{c}
{wiki}

A polynomial ring over a Noetherian ring is Noetherian.

= Noetherianity of a quadratic integer order
{parent=Noetherian ring}

For a nonsquare integer $d$,
$$
\mathbb Z[\sqrt d]\cong\mathbb Z[X]/(X^2-d).
$$
The integers are Noetherian, the Hilbert basis theorem makes $\mathbb Z[X]$ Noetherian, and quotients preserve Noetherianity.

= Nonprincipal ideal in the integers adjoined a square root of minus five
{parent=Noetherianity of a quadratic integer order}

In $R=\mathbb Z[\sqrt{-5}]$, the ideal
$$
I=(2,1+\sqrt{-5})
$$
has additive index two. If $I=(a+b\sqrt{-5})$ were principal, multiplication by its generator would give
$$
[R:I]=|a^2+5b^2|=2,
$$
which has no solution in integers. Thus $I$ is not principal.

= Constant-plus-ideal non-Noetherian subring
{parent=Noetherian ring}

The subring $R=K+XYK[X,Y]$ is not Noetherian. Its ideals
$$
(XY)_R\subsetneq(XY,XY^2)_R\subsetneq\cdots
$$
form a strict ascending chain because multiplying by a nonconstant element of $R$ introduces at least a second power of $X$.

= Pauli matrix
{parent=Algebra}
{c}
{wiki=Pauli_matrices}

= Pigeonhole principle
{parent=Algebra}
{wiki}

= Polynomial interpolation error
{parent=Algebra}
{wiki}

= Hermite interpolation
{parent=Polynomial interpolation error}
{c}
{wiki}

Hermite interpolation prescribes values and finitely many derivatives of a polynomial at distinct points.

= RSA cryptosystem
{parent=Algebra}
{c}
{wiki}

For $N=pq$, choose $e$ coprime to $\phi(N)$ and a private exponent $d$ with $ed\equiv1\pmod{\phi(N)}$. Encryption and decryption are
$$
c\equiv m^e\pmod N,
\qquad
m\equiv c^d\pmod N.
$$

= Common-modulus RSA attack
{parent=RSA cryptosystem}

If the same invertible message $m$ is sent under one RSA modulus with coprime public exponents $e_1,e_2$, choose integers $a,b$ with $ae_1+be_2=1$. From $c_i=m^{e_i}\pmod N$ one recovers
$$
m\equiv c_1^ac_2^b\pmod N,
$$
using modular inverses for negative exponents.

= RSA private exponent reveals the factorization
{parent=RSA cryptosystem}
{c}

Knowledge of a valid public and private exponent gives $ed-1$, a nonzero multiple of $\lambda(N)$. Writing it as $2^st$ with $t$ odd and testing random bases produces a nontrivial square root of one modulo $N$ with high probability; a greatest common divisor then factors $N$. Thus one customer's private key under a shared semiprime modulus compromises every exponent using that modulus.

= Root of unity
{parent=Algebra}
{wiki}

An $N$th root of unity is a <complex number> $z$ satisfying $z^N=1$. The primitive root $e^{2\pi i/N}$ generates all $N$th roots.

= Roots of unity
{synonym}

= Primitive root of unity
{parent=Root of unity}
{wiki=Root_of_unity#Primitive_nth_roots}

A primitive $n$th root of unity has multiplicative order exactly $n$.

= Root-of-unity filter
{parent=Root of unity}

For $\omega=e^{2\pi i/N}$,
$$
\sum_{y=0}^{N-1}\omega^{my}
=\begin{cases}N,&N\mid m,\\0,&N\nmid m.\end{cases}
$$
This is the finite <geometric series> identity and expresses orthogonality of the characters of $\mathbb Z_N$.

= Smith normal form
{parent=Algebra}
{c}
{wiki}

For every integer matrix $A$, there are unimodular matrices $U,V$ such that
$$
UAV=\operatorname{diag}(d_1,\ldots,d_r,0,\ldots,0),
\qquad d_1\mid d_2\mid\cdots\mid d_r.
$$
The positive diagonal entries are unique. Their products $d_1\cdots d_k$ are the greatest common divisors of the $k$-rowed minors.

= Star-shaped set
{parent=Algebra}
{wiki}

= Surjective function
{parent=Algebra}
{wiki}

= Transcendental number
{parent=Algebra}
{wiki}

= Liouville number
{parent=Transcendental number}
{c}
{wiki}

A Liouville number admits rational approximations of every fixed polynomial order; Liouville's approximation theorem makes every such irrational number transcendental.

= Unique factorization domain
{parent=Algebra}
{wiki}

= Unit in a ring
{parent=Algebra}
{wiki}

A unit in a ring is an element with a multiplicative inverse.

= Unit
{synonym}

= Unit group
{title2=$R^\times$}
{parent=Unit in a ring}
{wiki=Unit_(ring_theory)}

The units of a ring $R$ form a group $R^\times$ under multiplication.

= Multiplicative group of integers modulo n
{title2=$(\mathbb Z/n\mathbb Z)^\times$}
{parent=Unit group}

The multiplicative group of integers modulo $n$ consists of the residue classes coprime to $n$. It has order $\varphi(n)$.

= Wilson theorem
{parent=Algebra}
{c}
{wiki}

For a prime $p$,
$$
(p-1)!\equiv-1\pmod p.
$$
Every nonzero residue other than $1$ and $-1$ pairs with a distinct multiplicative inverse, leaving only $1(-1)$ in the product.

= Fundamental theorem of algebra
{parent=Algebra}
{wiki}

Every nonconstant complex polynomial has a complex root and therefore factors completely into linear factors over $\mathbb C$.

= Group theory
{parent=Algebra}
{wiki}

Group theory studies groups, homomorphisms, actions, quotients, and symmetry.

= Group
{parent=Group theory}
{wiki=Group_(mathematics)}

A group is a set with an associative binary operation, an identity, and an inverse for every element.

= Finite group
{parent=Group}
{wiki}

A finite group is a <group> with finitely many elements. Its number of elements is its order.

= Order of a finite group
{title2=$|G|$}
{parent=Finite group}

The order $|G|$ of a finite group is its number of elements.

= Nontrivial group
{parent=Group}

A nontrivial group has more than one element.

= Identity element
{parent=Group}
{wiki}

An identity element $e$ satisfies $eg=ge=g$ for every element $g$ of a <group>.

= Inverse element
{parent=Group}
{wiki=Inverse_element}

The inverse $g^{-1}$ of a group element $g$ satisfies $gg^{-1}=g^{-1}g=e$.

= Group operation
{parent=Group}
{wiki=Group_(mathematics)}

A group operation is the associative binary operation of a <group>. It has an <identity element>, and every element has an <inverse element>.

= Translation in a group
{parent=Group}
{wiki=Translation_(group_theory)}

Left and right multiplication by a fixed group element are bijections, called left and right translations.

= Cyclic group
{title2=$C_n$}
{parent=Group}
{wiki}

A cyclic group is generated by one element. A cyclic group of order $n$ has exactly $d$ solutions of $x^d=1$ for every positive <integer divisor> $d$ of $n$.

= Finite subgroup of the multiplicative complex numbers
{parent=Cyclic group}

Every finite subgroup of $\mathbb C^\times$ is cyclic. If its exponent is $n$, it is a subgroup of the cyclic group of $n$th roots of unity.

= Generator of a group
{parent=Cyclic group}
{wiki=Generating_set_of_a_group}

A generator of a <cyclic group> $G$ is an element $g$ such that every element of $G$ is an integer power of $g$.

= Abelian group
{parent=Group}
{wiki}

An abelian group is a <group> whose operation is commutative: $gh=hg$ for every $g,h$ in the group.

= Commutative group
{synonym}

= Finite additive group
{parent=Group}
{wiki=Group_(mathematics)}

A finite additive group is a <finite group> whose operation is written as <addition>. Its <identity element> is denoted $0$, the <inverse element> of $x$ is denoted $-x$, and <translation in a group>[translation] $y\mapsto x+y$ is a <bijection>.

= Subgroup
{parent=Group}
{wiki}

A subgroup is a subset of a <group> that is itself a group under the restricted operation.

= Index of a subgroup
{title2=$[G:H]$}
{parent=Subgroup}
{wiki=Index_of_a_subgroup}

The index $[G:H]$ is the number of left cosets of $H$ in $G$.

= Index-two subgroup is normal
{parent=Subgroup}

Every subgroup $H\leq G$ of index two is normal: the two left cosets and the two right cosets are both $H$ and its complement, so they agree.

= First isomorphism theorem
{parent=Group theory}
{wiki}

For a group homomorphism $\phi:G\to H$, the map $g\ker\phi\mapsto\phi(g)$ gives an isomorphism
$$
G/\ker\phi\cong\operatorname{im}\phi.
$$

= Direct product of groups
{parent=Group theory}
{wiki=Direct_product_of_groups}

The direct product $G\times H$ has componentwise multiplication. An element $(g,h)$ of finite component orders has order $\operatorname{lcm}(\operatorname{ord}g,\operatorname{ord}h)$.

= Commutator subgroup
{parent=Group theory}
{wiki}

The commutator subgroup $G'$ is generated by all $[x,y]=x^{-1}y^{-1}xy$. It is characteristic, $G/G'$ is abelian, and every normal subgroup with abelian quotient contains $G'$.

= Abelianization
{title2=$G^{\mathrm{ab}}$}
{parent=Commutator subgroup}
{wiki}

The abelianization of a group is the quotient $G^{\mathrm{ab}}=G/G'$ by its <commutator subgroup>. It is the largest abelian quotient of $G$.

= Center of a group
{parent=Group theory}
{wiki}

= Cyclic quotient by the center
{parent=Center of a group}

If $G/Z(G)$ is cyclic, then $G$ is abelian. Indeed, if $gZ(G)$ generates the quotient, every element is $g^az$ with $z\in Z(G)$, and any two such elements commute.

= Conjugate subgroup
{parent=Group theory}
{wiki}

= Group action
{parent=Group theory}
{wiki}

= Conjugation action
{parent=Group action}
{wiki=Group_action#Conjugation_action}

A group acts on itself and on its subgroups by conjugation: $g\cdot x=gxg^{-1}$ and $g\cdot H=gHg^{-1}$.

= Normalizer
{title2=$N_G(H)$}
{parent=Conjugation action}
{wiki=Centralizer_and_normalizer}

The normalizer of a subgroup $H\leq G$ is $N_G(H)=\{g:gHg^{-1}=H\}$.

= Orbit-stabilizer theorem
{parent=Group action}
{wiki}

For a finite group acting on a set and any point $x$,
$$
|G|=|Gx|\,|G_x|.
$$
Indeed, $gG_x\mapsto gx$ is a well-defined bijection from the left cosets of the stabilizer to the orbit.

= Doubly transitive group action
{parent=Group action}
{wiki=Multiply_transitive_group_action}

An action is doubly transitive when the group acts transitively on ordered pairs of distinct points. Equivalently, each point stabilizer acts transitively on the remaining points.

= Symmetry group of a cube
{parent=Group action}

The full isometry group of a cube has order $48$. Its transitive actions on the $8$ vertices, $12$ edges, and $4$ main diagonals have stabilizers isomorphic respectively to
$$
S_3,\qquad C_2\times C_2,\qquad C_2\times S_3.
$$
The action on main diagonals has kernel $\{I,-I\}$.

= Faithful group action
{parent=Group action}
{wiki}

A group action is faithful when only the identity acts as the identity on every point.

= Normal subgroup orbits in a faithful prime-degree action
{parent=Faithful group action}

If a group acts transitively on a set of prime size and $N$ is normal, then the $N$-orbits form a $G$-invariant block system of equal size. They are therefore either singletons or the whole set. In a faithful action the singleton case forces $N=1$, so every nontrivial normal subgroup is transitive.

= Orbit of a group action
{parent=Group action}
{wiki}

The orbit of $x$ is $Gx=\{gx:g\in G\}$.

= Stabilizer subgroup
{parent=Group action}
{wiki}

The stabilizer of $x$ is the subgroup $\{g:gx=x\}$.

= Symmetric group action on subsets
{parent=Group action}
{wiki}

The symmetric group acts on $k$-element subsets by applying a permutation to every member.

= Coset action
{parent=Group action}
{wiki}

A group acts on the left cosets of a subgroup by left multiplication; its kernel is the core of that subgroup.

= Coset
{parent=Group theory}
{wiki}

For a subgroup $H\leq G$, a left coset is $gH=\{gh:h\in H\}$. The left cosets partition $G$ and all have cardinality $|H|$.

= Double coset
{parent=Coset}
{wiki}

For subgroups $H,P\leq G$, the double coset of $x\in G$ is
$$
HxP=\{hxp:h\in H,\ p\in P\}.
$$
The double cosets partition $G$. The action of $H$ on the left cosets of $P$ gives
$$
\frac{|HxP|}{|P|}
=\frac{|H|}{|H\cap xPx^{-1}|}
$$
by the <orbit-stabilizer theorem>.

= Sylow subgroup of a subgroup from double cosets
{parent=Double coset}
{c}

If $P$ is a <Sylow subgroup> of a finite group $G$ and $H\leq G$, some intersection $H\cap xPx^{-1}$ is a Sylow subgroup of $H$. Otherwise every double-coset size
$$
|HxP|=\frac{|H||P|}{|H\cap xPx^{-1}|}
$$
would be divisible by a larger power of $p$ than $|G|$, as would their sum $|G|$.

= Lagrange's theorem
{parent=Group theory}
{c}
{wiki=Lagrange%27s_theorem_(group_theory)}

For a finite group $G$ and subgroup $H$, the order of $H$ divides the order of $G$, and the quotient is the number of left <coset>[cosets]:
$$
|G|=[G:H]|H|.
$$

= Lagrange theorem
{synonym}

= Group embedding
{parent=Group theory}
{wiki=Group_homomorphism\#Monomorphisms}

A group embedding is an injective group homomorphism. It identifies its domain with an isomorphic subgroup of its codomain.

= Cayley theorem
{parent=Group embedding}
{c}
{wiki=Cayley%27s_theorem}

Every group $G$ embeds in the symmetric group on its underlying set through the left-regular action $g:x\mapsto gx$.

= Symmetric-group embedding at one less than the group order
{parent=Cayley theorem}

For $n>1$, a cyclic group of prime-power order $n$ cannot embed in $S_{n-1}$, because a permutation of order $p^k$ requires a cycle of length at least $p^k$. If $n$ is not a prime power, every group of order $n$ embeds in $S_{n-1}$ by combining coset actions on subgroups of two distinct prime orders.

= Generalized dihedral group
{parent=Group theory}
{wiki=Generalized_dihedral_group}

For an abelian group $A$, the generalized dihedral group is the semidirect product $A\rtimes C_2$ in which the nonidentity element of $C_2$ acts on $A$ by inversion.

= Order of a group element
{parent=Group theory}
{wiki}

= Quotient group
{parent=Group theory}
{wiki}

= General linear group
{title2=$GL(V)$}
{parent=Group theory}
{wiki}

The general linear group $GL(V)$ is the <group> of invertible <linear map>[linear maps] from a <vector space> $V$ to itself, with composition as its operation.

= Special linear group
{title2=$SL(n)$}
{parent=Group theory}
{wiki}

= Group homomorphism
{parent=Group theory}
{wiki}

A group homomorphism preserves multiplication: $\phi(xy)=\phi(x)\phi(y)$.

= Homomorphism between cyclic groups
{parent=Group homomorphism}
{wiki}

A homomorphism from $C_n$ is determined by the image of one generator; a surjection $C_n\to C_m$ exists exactly when $m\mid n$.

= Kernel of a group homomorphism
{parent=Group homomorphism}
{wiki}

The kernel $\{g:\phi(g)=1\}$ is a normal subgroup of the domain.

= Image of a group homomorphism
{parent=Group homomorphism}
{wiki=Group_homomorphism}

The image of a group homomorphism $\phi:G\to H$ is the subgroup $\{\phi(g):g\in G\}$ of $H$.

= Automorphism group
{title2=$\operatorname{Aut}(G)$}
{parent=Group theory}
{wiki}

The automorphism group of a <group> $G$ is the group of all isomorphisms $G\to G$ under composition.

= Finite group theory
{parent=Group theory}
{wiki}

Finite group theory studies groups with finitely many elements.

= Alternating group
{title2=$A_n$}
{parent=Finite group theory}
{wiki}

= P-group
{parent=Finite group theory}
{wiki}

= Symmetric group
{title2=$S_n$}
{parent=Finite group theory}
{wiki}

= Sign homomorphism
{title2=$\operatorname{sgn}$}
{parent=Symmetric group}
{wiki=Parity_of_a_permutation}

The sign homomorphism sends an even permutation to $1$ and an odd permutation to $-1$; its kernel is the <alternating group>.

A symmetric group is the <group> of all <bijections> from a set to itself, with <function composition> as its operation.

= Permutation cycle
{parent=Symmetric group}
{wiki=Cycle_(permutation)}

A permutation cycle $(a_1\,a_2\,\ldots\,a_k)$ sends $a_i$ to $a_{i+1}$, sends $a_k$ to $a_1$, and fixes every other element. Its <order of a group element>[order] is $k$.

= Three-cycle
{parent=Permutation cycle}
{wiki=Cycle_(permutation)}

A three-cycle is a <permutation cycle> of length three. On any chosen set of three letters, the two possible three-cycles are inverses of one another.

= Disjoint permutation cycles
{parent=Permutation cycle}
{wiki=Permutation#Cycle_notation}

Permutation cycles with disjoint supports commute. The order of their product is the <least common multiple> of their lengths.

= Support of a permutation
{parent=Symmetric group}
{wiki=Support_(mathematics)}

The support of a permutation is the set of elements it does not fix. A permutation has finite support when this set is finite.

= Sign of a permutation
{parent=Symmetric group}
{wiki=Parity_of_a_permutation}

The sign homomorphism $\operatorname{sgn}:S_n\to\{\pm1\}$ is the determinant of the corresponding permutation matrix. A transposition has sign $-1$, so this also proves that the parity of any transposition decomposition is well-defined.

= Quaternion group
{parent=Finite group theory}
{wiki}

The quaternion group $Q_8=\{\pm1,\pm i,\pm j,\pm k\}$ is a nonabelian Dedekind group.

= Dicyclic group
{parent=Finite group theory}
{wiki}

The dicyclic group of order $4n$ has generators
$$
a^{2n}=1,\qquad b^2=a^n,\qquad bab^{-1}=a^{-1}.
$$
Every element has a unique form $a^k$ or $a^kb$, with $0\leq k<2n$.

= Simple group
{parent=Finite group theory}
{wiki}

A nontrivial simple group has no normal subgroups other than the identity subgroup and itself.

= Simplicity of the alternating group A5
{parent=Simple group}
{c}

The conjugacy classes of $A_5$ have sizes
$$
1,\quad15,\quad20,\quad12,\quad12.
$$
A normal subgroup is a union of these classes containing the identity. No proper nontrivial such union has size dividing $60$, so Lagrange's theorem proves that $A_5$ is simple.

= Smallest nonabelian simple group
{parent=Simple group}
{wiki}

Every finite nonabelian simple group has order at least 60; the alternating group A5 attains this bound.

= General linear group over a finite field
{parent=Finite group theory}
{wiki}

The group $GL_n(\mathbb F_q)$ consists of invertible $n$ by $n$ matrices over the finite field $\mathbb F_q$.

= Order of a general linear group over a finite field
{parent=General linear group over a finite field}

Choosing linearly independent columns successively gives
$$
|GL_n(\mathbb F_q)|
=(q^n-1)(q^n-q)\cdots(q^n-q^{n-1})
=q^{n(n-1)/2}\prod_{k=1}^n(q^k-1).
$$

= Upper unitriangular group
{parent=General linear group over a finite field}
{wiki=Unitriangular_matrix}

The upper unitriangular group consists of upper triangular matrices with every diagonal entry equal to one. Over $\mathbb F_p$ it has order $p^{n(n-1)/2}$ and is a <Sylow subgroup> of $GL_n(\mathbb F_p)$.

= Projective general linear group action on the projective line
{parent=General linear group over a finite field}
{wiki=Projective_linear_group}

The group $PGL_2(F)=GL_2(F)/Z$ acts faithfully on
$$
\mathbb P^1(F)=F\cup\{\infty\}
$$
by <Möbius transformations>. For $|F|=q$, this embeds $PGL_2(F)$ into the <symmetric group> $S_{q+1}$.

= Projective line
{parent=Projective general linear group action on the projective line}
{wiki}

The projective line over a field $F$ is the set of one-dimensional subspaces of $F^2$ and may be identified with $F\cup\{\infty\}$.

= Sylow 2-subgroup of PGL2 over F4
{parent=Projective general linear group action on the projective line}
{c}

The translations $x\mapsto x+b$ for $b\in\mathbb F_4$ form a Sylow $2$-subgroup of $PGL_2(\mathbb F_4)$. Every nonidentity translation fixes infinity and exchanges the four finite points in two transpositions, so its action lies in $A_5$.

= Special linear group over a finite field
{parent=General linear group over a finite field}
{wiki}

The determinant-one matrices form the normal subgroup $SL_n(\mathbb F_q)=\ker(\det)$.

= Dihedral group
{title2=$D_n$}
{parent=Finite group theory}
{wiki}

The dihedral group $D_{2n}$ is generated by a rotation $r$ of order $n$ and a reflection $s$ with $srs=r^{-1}$; it has order $2n$.

= Sylow theorems
{parent=Finite group theory}
{c}
{wiki}

If $|G|=p^am$ with $p\nmid m$, then $G$ has subgroups of order $p^a$, every $p$-subgroup lies in one, and all such Sylow subgroups are conjugate. Their number satisfies $n_p\equiv1\pmod p$ and $n_p\mid m$.

= Sylow theorem
{c}
{synonym}

= Sylow subgroup
{c}
{parent=Sylow theorems}
{wiki}

A Sylow $p$-subgroup of a finite <group> has the largest power of $p$ dividing the group order.

= Nonabelian group of order pq
{parent=Sylow theorems}

Let $p>q$ be <prime numbers>. A nonabelian group of order $pq$ has a unique Sylow $p$-subgroup and exactly $p$ Sylow $q$-subgroups. Hence the Sylow count satisfies $p\equiv1\pmod q$, so $q\mid p-1$.

= Conjugation action on Sylow subgroups
{parent=Sylow theorems}
{wiki}

A group, or one of its Sylow subgroups, acts on the set of Sylow subgroups by conjugation.

= Sylow subgroups of S3, S4 and A5
{parent=Sylow theorems}
{c}

The Sylow $2$-subgroups of $S_3$ are its three transposition subgroups, while its unique Sylow $3$-subgroup is $A_3$. The three Sylow $2$-subgroups of $S_4$ are dihedral groups of order eight and are the normalizers of the three cyclic subgroups generated by inverse pairs of $4$-cycles. The five Sylow $2$-subgroups of $A_5$ are Klein four-groups, one fixing each letter.

= Sylow containment from a coset fixed point
{parent=Sylow theorems}
{c}

If $P$ is a Sylow $p$-subgroup of $G$ and $Q$ is any $p$-subgroup, let $Q$ act on $G/P$. Since $p$ does not divide $|G/P|$, some coset $gP$ is fixed. Hence $g^{-1}Qg\leq P$, or equivalently $Q\leq gPg^{-1}$.

= Sylow counts in a faithful degree-seven action with S4 point stabilizers
{parent=Sylow theorems}
{c}

Suppose a group acts faithfully and transitively on seven points, every point stabilizer is $S_4$, and every two-point stabilizer is a Klein four-group. Then the group has order $168$ and
$$
n_2=21,
\qquad n_3=28,
\qquad n_7=8.
$$
Count pairs consisting of a point and a Sylow subgroup fixing it for $p=2,3$; the two-point stabilizer ensures uniqueness. For $p=7$, a normal Sylow subgroup would force the group into its order-$42$ normalizer in $S_7$.

= Even-involution coset fixed-point lemma
{parent=Sylow theorems}

Suppose a finite group $G$ has no index-two subgroup, $P$ is a Sylow $2$-subgroup, $H$ has index two in $P$, and $x\in G$ has order two. The sign of the action on $G/H$ defines a homomorphism $G\to\{\pm1\}$ and is therefore trivial. Since $[G:H]\equiv2\pmod4$, a fixed-point-free involution would be an odd number of transpositions. Thus $x$ fixes a coset $gH$, equivalently $g^{-1}xg\in H$.

= Cauchy theorem for groups
{parent=Finite group theory}
{c}
{wiki}

If a prime divides the order of a finite group, the group contains an element of that prime order.

One proof lets a cyclic group of order $p$ rotate the tuples
$$
(x_1,\ldots,x_p)\quad\hbox{with}\quad x_1\cdots x_p=1.
$$
The tuple set has size $|G|^{p-1}$, and its fixed tuples are precisely $(x,\ldots,x)$ with $x^p=1$. Counting nonfixed orbits modulo $p$ forces a nonidentity fixed tuple.

= Composition series
{parent=Finite group theory}
{wiki}

A composition series is a finite subnormal chain whose successive quotients are simple.

= Klein four-group
{parent=Finite group theory}
{c}
{wiki}

The Klein four-group is the abelian group with three nonidentity elements, all of order two.

= Torsion element
{parent=Group theory}
{wiki}

A torsion element has finite order.

= Free abelian group
{parent=Group theory}
{wiki}

A free abelian group has a basis over $\mathbb Z$ and is isomorphic to a direct sum of copies of $\mathbb Z$.

= Normal subgroup
{parent=Group theory}
{wiki}

A subgroup $N\le G$ is normal when $gNg^{-1}=N$ for every $g\in G$.

= Preimage of a normal subgroup
{parent=Normal subgroup}
{wiki}

The preimage of a normal subgroup under any group homomorphism is normal.

= Image of a normal subgroup
{parent=Normal subgroup}
{wiki}

Under a surjective homomorphism, the image of a normal subgroup is normal.

= Dedekind group
{parent=Group theory}
{c}
{wiki}

A Dedekind group is a group in which every subgroup is normal.

= Projective linear group
{parent=Group theory}
{wiki}

The projective linear group is $PGL_n(F)=GL_n(F)/F^\times$ and acts on projective space.

= Möbius transformation
{parent=Projective linear group}
{c}
{wiki}

A Möbius transformation of the Riemann sphere has the form
$$
z\longmapsto\frac{az+b}{cz+d},
\qquad ad-bc\ne0.
$$
The group is generated by translations, nonzero complex scalings, and inversion $z\mapsto1/z$.

= Cross-ratio
{parent=Möbius transformation}
{wiki=Cross-ratio}

For four distinct points of the <Riemann sphere>, their cross-ratio is
$$
[z_1,z_2,z_3,z_4]
=\frac{(z_4-z_2)(z_1-z_3)}
{(z_4-z_3)(z_1-z_2)},
$$
with the evident limiting conventions at infinity. Every Möbius transformation preserves it.

= Generalized circle under a Möbius transformation
{parent=Möbius transformation}
{c}

A generalized circle is either a Euclidean circle or a straight line together with the point at infinity. Every Möbius transformation maps generalized circles to generalized circles. A Euclidean circle becomes a line exactly when it contains the pole, the preimage of infinity.

= Affine subgroup of the Möbius group
{parent=Möbius transformation}
{c}

The transformations that take every Euclidean circle to a Euclidean circle are precisely
$$
z\longmapsto az+b,\qquad a\ne0.
$$
They form the stabilizer of infinity in the Möbius group and are not a normal subgroup.

= Fixed point of a Möbius transformation
{parent=Möbius transformation}
{wiki}

A fixed point of a Möbius transformation satisfies a quadratic equation on the Riemann sphere.

= Prescribed fixed points of a Möbius transformation
{parent=Fixed point of a Möbius transformation}

For $f(z)=(az+b)/(z+1)$, the finite fixed points are the roots of
$$
z^2+(1-a)z-b=0.
$$
Thus a repeated fixed point or two prescribed fixed points determine $a$ and $b$ by the usual sum and product of roots.

= Fixed points of a finite-order Möbius transformation
{parent=Fixed point of a Möbius transformation}

Every nonidentity finite-order Möbius transformation has exactly two fixed points on the Riemann sphere. A transformation with one repeated fixed point is conjugate to a nontrivial translation and therefore has infinite order.

= Constant-argument locus of a Möbius transformation
{parent=Möbius transformation}

A condition $\arg((z-z_1)/(z-z_2))=\theta$ fixes the oriented angle subtended by $z_1,z_2$. Its locus is an arc of a circle through those two points, with the endpoints excluded.

= Classification of Möbius transformations by trace
{parent=Möbius transformation}
{wiki}

For determinant-one real representatives, trace magnitude below, equal to, or above two gives elliptic, parabolic, or hyperbolic type.

= Möbius maps commuting with reflection in the unit circle
{parent=Möbius transformation}
{c}

The Möbius transformations commuting with $J(z)=1/\overline z$ are exactly
$$
z\longmapsto\frac{az+b}{\overline bz+\overline a},
\qquad |a|^2-|b|^2\ne0.
$$
They preserve the open unit disc exactly when $|a|>|b|$. The identity
$$
|az+b|^2-|\overline bz+\overline a|^2
=(|a|^2-|b|^2)(|z|^2-1)
$$
proves the latter assertion.

= Matrix representative of a Möbius transformation
{parent=Projective linear group}
{wiki}

A Möbius transformation is represented by an invertible two-by-two matrix, with nonzero scalar multiples representing the same map.

= Semidirect product
{parent=Group theory}
{wiki}

A semidirect product combines a normal subgroup with a subgroup acting on it by automorphisms.

= Affine group over a finite field
{parent=Semidirect product}
{wiki}

The one-dimensional affine group over $\mathbb F_q$ consists of maps $x\mapsto ax+b$ with $a\ne0$.

= Affine Galois group of the splitting field of x to the p minus two
{parent=Affine group over a finite field}

For an odd prime $p$, the splitting field
$$
\mathbb Q(\zeta_p,2^{1/p})
$$
of $X^p-2$ has Galois group
$$
C_p\rtimes\mathbb F_p^\times
\cong\operatorname{AGL}_1(\mathbb F_p).
$$
The translation subgroup multiplies $2^{1/p}$ by powers of $\zeta_p$, while the multiplicative subgroup acts on $\zeta_p$ by cyclotomic automorphisms.

= Affine semidirect product of cyclic groups of orders eleven and five
{parent=Affine group over a finite field}

Let $Q$ be the subgroup of nonzero squares in $\mathbb F_{11}^\times$. The affine maps
$$
x\longmapsto rx+b,
\qquad r\in Q,quad b\in\mathbb F_{11},
$$
form the nonabelian semidirect product $C_{11}\rtimes C_5$, with multiplication
$$
(r,b)(s,c)=(rs,b+rc).
$$

= Conjugacy classes in the affine semidirect product of orders eleven and five
{parent=Affine semidirect product of cyclic groups of orders eleven and five}

This group has seven conjugacy classes: the identity; two classes of five nonidentity translations, distinguished by square class of the translation parameter; and, for each of the four $r\in Q\setminus\{1\}$, one class $\{(r,b):b\in\mathbb F_{11}\}$ of size eleven.

= Solvable group
{parent=Group theory}
{wiki}

A solvable group has a subnormal series whose factor groups are abelian.

= Exponent of a finite group
{parent=Group theory}
{wiki=Exponent_of_a_group}

The exponent of a finite group is the least common multiple of the orders of all its elements.

= Finite subgroup of a field multiplicative group is cyclic
{parent=Exponent of a finite group}

Every finite subgroup of $K^\times$ is cyclic. If its exponent is $m$, all its elements are roots of $X^m-1$, so its size is at most $m$; finite abelian group theory supplies an element of order $m$, forcing equality.

= Order of an element of a finite group
{parent=Group theory}
{wiki=Order_(group_theory)}

The order of an element is the least positive power equal to the identity and divides the order of a finite group.

= Homogeneous polynomial
{parent=Algebra}
{wiki}

A polynomial is homogeneous of degree $n$ when scaling all variables by $\lambda$ scales its value by $\lambda^n$.

= Commutative algebra
{parent=Algebra}
{wiki}

Commutative algebra studies commutative rings, ideals, modules, and their spectra.

= Polynomial ring
{parent=Commutative algebra}
{wiki}

The polynomial ring $R[X]$ consists of polynomials in an indeterminate $X$ with coefficients in a ring $R$.

= Maximal ideal
{parent=Commutative algebra}
{wiki}

A proper ideal $\mathfrak m$ of a commutative ring $R$ is maximal when no proper ideal lies strictly between $\mathfrak m$ and $R$. Equivalently, $R/\mathfrak m$ is a field.

= Maximal ideal quotient criterion
{parent=Maximal ideal}

An ideal $\mathfrak m$ of a commutative unital ring $R$ is maximal exactly when $R/\mathfrak m$ is a field.

= Local ring
{parent=Commutative algebra}
{wiki=Local_ring}

A local ring is a <commutative ring> with exactly one maximal ideal.

= Discrete valuation ring
{parent=Local ring}
{wiki=Discrete_valuation_ring}

A discrete valuation ring is a <principal ideal domain> with exactly one nonzero maximal ideal. Every nonzero element of its <fraction field> is a unit times a unique integer power of a uniformizer.

= Ring
{parent=Commutative algebra}
{wiki=Ring_(mathematics)}

A ring has addition and multiplication, with an abelian group under addition and multiplication distributive over addition.

= Nilpotent element
{parent=Ring}
{wiki=Nilpotent}

An element $x$ of a ring is nilpotent when $x^n=0$ for some positive integer $n$. A field has no nonzero nilpotent elements.

= Commutative ring
{parent=Ring}
{wiki}

A commutative ring is a <ring> whose multiplication satisfies $ab=ba$.

= Graded ring
{parent=Ring}
{wiki}

A graded ring is a direct sum $R=\bigoplus_{n\geq0}R_n$ with $R_mR_n\subseteq R_{m+n}$.

= Ideal
{parent=Ring}
{wiki=Ideal_(ring_theory)}

An ideal is an additive subgroup of a <ring> that absorbs multiplication by ring elements.

= Generating set of an ideal
{parent=Ideal}

A set $S\subseteq I$ generates an ideal $I$ when every element of $I$ is a finite sum of ring multiples of elements of $S$.

= Principal ideal
{title2=$(a)$}
{parent=Ideal}
{wiki}

A principal ideal is generated by one element: $(a)=\{ra:r\in R\}$.

= Product of ideals
{title2=$IJ$}
{parent=Ideal}
{wiki=Ideal_(ring_theory)#Product}

The product $IJ$ consists of finite sums of products $xy$ with $x\in I$ and $y\in J$.

= Intersection of ideals
{title2=$I\cap J$}
{parent=Ideal}

The set-theoretic intersection of any family of ideals is again an ideal.

= Comaximal ideals
{parent=Ideal}
{wiki=Coprime_ideals}

Ideals $I$ and $J$ are comaximal when $I+J=R$.

= Prime-power ideal
{parent=Ideal}

A prime-power ideal has the form $\mathfrak p^m$ for a <prime ideal> $\mathfrak p$ and positive integer $m$.

= Prime-power ideals
{synonym}

= Ideal approximation theorem
{parent=Ideal}

For finitely many distinct prime ideals, one can prescribe finite valuation data simultaneously by the <Chinese remainder theorem>.

= Reduction modulo an ideal
{parent=Ideal}

Reduction modulo an ideal $I$ is the quotient homomorphism $R\to R/I$.

= Homogeneous ideal
{parent=Ideal}
{wiki=Homogeneous_ideal}

An <ideal> in a <graded ring> is homogeneous when it is generated by <homogeneous polynomial>[homogeneous elements], equivalently when every homogeneous component of each of its elements also belongs to the ideal.

= Radical of an ideal
{title2=$\sqrt I$}
{parent=Ideal}
{wiki=Radical_of_an_ideal}

The radical of an <ideal> $I$ is
$$
\sqrt I=\{f:f^m\in I\text{ for some }m\geq1\}.
$$
An ideal equal to its radical is a radical ideal.

= Ring homomorphism
{parent=Ring}
{wiki}

A ring homomorphism preserves addition, multiplication, and, under the convention used here, the multiplicative identity.

= Evaluation homomorphism
{parent=Ring homomorphism}
{wiki=Polynomial_ring\#Polynomial_evaluation}

For an element $a$ of a commutative $R$-algebra, evaluation is the ring homomorphism
$$
R[X]\longrightarrow R[a],\qquad f\longmapsto f(a).
$$

= First isomorphism theorem for rings
{parent=Ring homomorphism}
{wiki=Isomorphism_theorems\#Rings}

Every ring homomorphism $\phi:R\to S$ induces an isomorphism
$$
R/\ker\phi\cong\operatorname{im}\phi,
\qquad r+\ker\phi\longmapsto\phi(r).
$$

= Idempotent
{parent=Ring}
{wiki=Idempotent_(ring_theory)}

An element $e$ is idempotent when $e^2=e$.

= Integral domain
{parent=Commutative algebra}
{wiki}

An integral domain is a nonzero commutative ring with identity and without zero divisors.

= Prime element
{parent=Integral domain}
{wiki}

A nonzero nonunit $p$ in an integral domain is prime when $p\mid ab$ implies $p\mid a$ or $p\mid b$.

= Irreducible element
{parent=Integral domain}
{wiki}

A nonzero nonunit $p$ in an integral domain is irreducible when every factorization $p=ab$ has at least one unit factor.

= Greatest-common-divisor domain
{parent=Integral domain}
{wiki=GCD_domain}

A greatest-common-divisor domain is an integral domain in which every two nonzero elements have a greatest common divisor, defined up to multiplication by a unit.

= GCD domain
{c}
{synonym}

= Euclid lemma in a greatest-common-divisor domain
{parent=Greatest-common-divisor domain}
{c}

If $\gcd(a,b)=1$ and $a\mid bc$ in a greatest-common-divisor domain, then $a\mid c$. Indeed, the least common multiple of $a$ and $b$ is associate to $ab$ and divides every common multiple; applying this to $bc$ and cancelling $b$ proves the claim.

= Fraction field
{parent=Integral domain}
{wiki=Field_of_fractions}

The fraction field of an <integral domain> $R$ consists of fractions $a/b$ with $a,b\in R$ and $b\ne0$, modulo the usual equivalence relation.

= Principal ideal domain
{parent=Integral domain}
{wiki}

A principal ideal domain is an integral domain whose every ideal has one generator.

= Submodule theorem for free modules over a principal ideal domain
{parent=Principal ideal domain}

Every submodule of a finite-rank free module over a principal ideal domain is free. Induction on rank splits off a generator of the image under one coordinate projection and applies the result to the kernel.

= Euclidean domain
{parent=Principal ideal domain}
{wiki}

A Euclidean domain admits division with remainder measured by a decreasing Euclidean function.

= Euclidean function
{parent=Euclidean domain}
{wiki}

A Euclidean function on an integral domain $R$ assigns a nonnegative integer to every nonzero element so that, for $a,b\in R$ with $b\ne0$, there are $q,r\in R$ satisfying $a=bq+r$ and either $r=0$ or $d(r)<d(b)$.

= Gaussian integer
{title2=$\mathbb Z[i]$}
{c}
{parent=Euclidean domain}
{wiki}

A Gaussian integer has the form $a+bi$ with $a,b\in\mathbb Z$. Its multiplicative norm is $N(a+bi)=a^2+b^2$, which makes $\mathbb Z[i]$ a Euclidean domain.

= Gaussian integers
{c}
{synonym}

= Eisenstein integer
{parent=Euclidean domain}
{c}
{wiki}

For a primitive cube root of unity $\omega$, the Eisenstein integers form the triangular lattice
$$
\mathbb Z[\omega]=\{a+b\omega:a,b\in\mathbb Z\}.
$$
Their multiplicative norm is $N(a+b\omega)=a^2-ab+b^2$.

= Eisenstein integers
{synonym}

= Euclidean norm on the Eisenstein integers
{parent=Eisenstein integer}
{c}

Every complex number is within squared distance at most $3/4$ of an Eisenstein integer: write it as $x+y\omega$ and round $x$ and $y$ to integers. Therefore, for nonzero $\beta$ and arbitrary $\alpha$, a nearest lattice point $q$ to $\alpha/\beta$ gives
$$
N(\alpha-q\beta)<N(\beta).
$$
The norm makes the Eisenstein integers a <Euclidean domain>.

= Bézout domain
{parent=Integral domain}
{c}
{wiki=Bezout_domain}

A Bézout domain is an integral domain in which every finitely generated ideal is principal. Equivalently, every pair $a,b$ has a greatest common divisor $d$ satisfying a Bézout identity $d=ra+sb$. A Noetherian Bézout domain is a principal ideal domain because all its ideals are finitely generated.

= Ring product decomposition by an idempotent
{parent=Commutative algebra}

A commutative ring $R$ has a nontrivial idempotent $e$ exactly when it decomposes as a product of two nontrivial rings. The isomorphism determined by $e$ is
$$
R\longrightarrow eR\times(1-e)R,
\qquad r\longmapsto(er,(1-e)r),
$$
with inverse $(x,y)\mapsto x+y$.

= Fiber product of rings
{parent=Commutative algebra}
{wiki}

A fiber product of rings consists of pairs having the same image in a third ring.

= Unit modulo a prime power
{parent=Commutative algebra}
{wiki}

Modulo a prime power, a class is a unit exactly when it is not divisible by that prime.

= Nilpotent element modulo a prime power
{parent=Commutative algebra}
{wiki}

Modulo p^e, a class is nilpotent exactly when its reduction modulo p is zero.

= Reduction map on unit groups
{parent=Commutative algebra}
{wiki}

Reduction between quotient rings sends units to units and is surjective when every target unit avoids the same prime factors.

= Polynomial content
{parent=Commutative algebra}
{wiki}

The content of a polynomial is a greatest common divisor of its coefficients, up to a unit.

= Primitive polynomial
{parent=Polynomial content}
{wiki}

A primitive polynomial has unit content.

= Primitive
{synonym}

= Gauss lemma for polynomials
{parent=Primitive polynomial}
{c}
{wiki}

Over a UFD, products of primitive polynomials are primitive and factorization over the fraction field descends after content is removed.

= Polynomial ring over a unique factorization domain
{parent=Gauss lemma for polynomials}

If $R$ is a <unique factorization domain> with fraction field $F$, then $R[X]$ is a unique factorization domain. A primitive polynomial irreducible in $R[X]$ remains irreducible, hence prime, in the principal ideal domain $F[X]$; <Gauss lemma for polynomials> brings divisibility back to $R[X]$.

= Symmetric rational function in two variables
{parent=Gauss lemma for polynomials}

Every rational function in $\mathbb C(X,Y)$ fixed by exchanging $X$ and $Y$ is a quotient of coprime symmetric polynomials. In a coprime presentation $p/q$, symmetry gives
$$
pq^*=p^*q.
$$
Unique factorization implies $p^*=\lambda p$ and $q^*=\lambda q$. The alternative $\lambda=-1$ would make both divisible by $X-Y$, contradicting coprimality, so $\lambda=1$.

= Eisenstein criterion
{parent=Commutative algebra}
{c}
{wiki}

For a primitive polynomial
$$
F(X)=a_nX^n+\cdots+a_0\in\mathbb Z[X],
$$
if a prime $p$ does not divide $a_n$, divides every $a_j$ for $j<n$, and has $p^2\nmid a_0$, then $F$ is irreducible. If $F=GH$, reduction modulo $p$ forces both $\overline G$ and $\overline H$ to be positive-degree monomials. Their constant terms are consequently divisible by $p$, making $p^2\mid F(0)$, a contradiction. <Gauss lemma for polynomials> transfers the result between $\mathbb Z[X]$ and $\mathbb Q[X]$.

= Geometric-sum irreducibility criterion
{parent=Eisenstein criterion}

For $n>1$,
$$
1+X+\cdots+X^{n-1}
$$
is irreducible over the integers exactly when $n$ is prime. Composite $n$ gives a geometric factorization; for prime $p$, translation $X\mapsto X+1$ makes the polynomial Eisenstein at $p$.

= Odd-valuation obstruction to a rational-function square
{parent=Commutative algebra}

Every zero and pole of a square in $K(Y)$ has even order. Thus a rational function with a zero or pole of odd order cannot be a square. For characteristic different from two, this proves that $1-Y^2$ is not a square in $K(Y)$.

= Module theory
{parent=Commutative algebra}
{wiki=Module_(mathematics)}

Module theory extends linear algebra by allowing scalars from a ring rather than only a field.

= Primary decomposition theorem for finitely generated modules over a principal ideal domain
{parent=Module theory}
{wiki=Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain}

A finitely generated module over a principal ideal domain decomposes as
$$
R^r\oplus\bigoplus_p\bigoplus_jR/(p^{e_{p,j}}),
$$
with finitely many irreducibles $p$. The free rank and primary cyclic factors are unique up to associates and ordering.

= Module over a ring
{parent=Module theory}
{wiki=Module_(mathematics)}

An $R$-module is an abelian group equipped with scalar multiplication by a ring $R$, satisfying the usual distributive and associative laws.

= Submodule
{parent=Module over a ring}
{wiki=Submodule}

A submodule $N$ of an $R$-module $M$ is a subset closed under addition, additive inverses, and scalar multiplication by every element of $R$.

= R-module
{c}
{synonym}

= Quotient module
{parent=Module theory}
{wiki=Quotient_module}

For a submodule $N\leq M$, the quotient module $M/N$ consists of additive cosets with scalar multiplication $r(m+N)=rm+N$.

= Free module
{parent=Module theory}
{wiki}

A free module has a basis: every element has a unique finite linear combination in that basis with coefficients in the scalar ring.

= Basis of a module
{parent=Free module}

A basis $S$ of an $R$-module $M$ is a subset such that every element of $M$ has a unique expression as a finite $R$-linear combination of elements of $S$.

= Universal property of a free module
{parent=Free module}

A module $F$ is free on a set $S$ exactly when every function from $S$ to any module $N$ extends uniquely to an $R$-module homomorphism $F\to N$.

= Rank of a free module
{title2=$\operatorname{rk}M$}
{parent=Free module}

For a nonzero commutative scalar ring, every two bases of a finite-rank free module have the same cardinality. This cardinality is the rank.

= Invariant basis number for a commutative ring
{parent=Free module}
{wiki=Invariant_basis_number}

If $R$ is a nonzero commutative ring and $R^m\cong R^n$, reduction modulo a maximal ideal produces isomorphic vector spaces of dimensions $m$ and $n$. Hence $m=n$.

= Torsion-free module
{parent=Module theory}
{wiki}

A module $M$ over an integral domain $R$ is torsion-free when $rm=0$ with $0\ne r\in R$ implies $m=0$.

= Projective module
{parent=Module theory}
{wiki}

A module $P$ is projective when every map from $P$ through the target of a surjection lifts through that surjection.

= Free modules are projective
{parent=Projective module}

Lift the image of each basis element independently through the given surjection, then extend the chosen lifts linearly.

= Cyclic module
{parent=Module theory}
{wiki}

A module is cyclic when one element $m$ generates it: $M=Rm$.

= Irreducible module
{parent=Module theory}
{wiki=Simple_module}

An irreducible, or simple, module is nonzero and has no submodules other than zero and itself.

= Simple module over a commutative ring
{parent=Irreducible module}

Every simple module over a commutative ring $R$ is isomorphic to $R/\mathfrak m$ for a maximal ideal $\mathfrak m$; its annihilator is therefore maximal.

= Annihilator of a module
{parent=Module theory}
{wiki=Annihilator_(ring_theory)}

The annihilator $\operatorname{Ann}_R(M)$ is the ideal of scalars that kill every element of $M$.

= Structure theorem for finitely generated modules over a principal ideal domain
{parent=Module theory}
{wiki=Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain}

Every finitely generated module over a PID is a direct sum of a free module and cyclic prime-power torsion modules.

= Finitely generated torsion-free module over a principal ideal domain
{parent=Structure theorem for finitely generated modules over a principal ideal domain}

Every finitely generated torsion-free module over a principal ideal domain is free. In the structure theorem, torsion-freeness removes every cyclic torsion summand and leaves only the free summand.

= Classification of integral matrices satisfying the third cyclotomic polynomial
{parent=Finitely generated torsion-free module over a principal ideal domain}

An integral matrix satisfying $A^2+A+I=0$ makes $\mathbb Z^n$ a module over the <Eisenstein integers> by letting the primitive cube root act as $A$. This module is finitely generated and torsion-free, hence free. Since the Eisenstein integers have integer rank two, such matrices exist only for $n=2m$; for that dimension every one is integrally conjugate to
$$
\begin{pmatrix}0&-1\\1&-1\end{pmatrix}^{\oplus m}.
$$

= Elementary divisor
{parent=Structure theorem for finitely generated modules over a principal ideal domain}
{wiki}

An elementary divisor is a prime power $p^e$ occurring in the cyclic decomposition of a finitely generated torsion module.

= Indecomposable finite abelian groups
{parent=Structure theorem for finitely generated modules over a principal ideal domain}

A finite abelian group is indecomposable exactly when it is cyclic of prime-power order. The structure theorem proves necessity. Conversely, every nonzero subgroup of $C_{p^n}$ contains its unique subgroup of order $p$, so two nonzero subgroups cannot be complementary direct summands.

= Prime ideal
{parent=Commutative algebra}
{wiki}

An ideal $\mathfrak p$ is prime when $ab\in\mathfrak p$ implies $a\in\mathfrak p$ or $b\in\mathfrak p$, equivalently when the quotient is an integral domain.

= Prime ideal quotient criterion
{parent=Prime ideal}

For a commutative unital ring $R$, an ideal $I$ is prime exactly when $R/I$ is an integral domain. The zero-product condition in the quotient translates directly to $ab\in I$.

= Boolean ring
{parent=Commutative algebra}
{wiki}

A Boolean ring satisfies $r^2=r$ for every element. Expanding $(r+1)^2=r+1$ shows that it has characteristic two.

= Prime ideals of a Boolean ring are maximal
{parent=Boolean ring}

A nonzero Boolean integral domain has only the elements zero and one, so it is $\mathbb F_2$. The quotient criterion then makes every prime quotient of a Boolean ring a field, and every prime ideal maximal.

= Coefficientwise quotient of a polynomial ring
{parent=Commutative algebra}

For an ideal $I$ of a commutative ring $R$, reducing coefficients modulo $I$ is a surjection $R[X]\to(R/I)[X]$ with kernel $I[X]$. Hence
$$
R[X]/I[X]\cong(R/I)[X].
$$

= Irreducible real polynomial
{parent=Commutative algebra}

The irreducible polynomials over $\mathbb R$ have degree one, or degree two with negative discriminant.

= Kummer irreducibility criterion
{parent=Commutative algebra}
{c}

Over a field containing the relevant roots of unity, $X^n-a$ is irreducible when the valuation data prevent $a$ from being a proper prime-divisor power, with the standard fourth-power exception.