Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 18F / a / Solution

Define ${\Phi }_n(X)={\prod }_{1\le k\le n,(k,n)=1}(X-e^{2\pi ik/n})$. From $X^n-1={\prod }_{d\mid n}{\Phi }_d(X)$, induction and monic division show ${\Phi }_n\in \mathbb{Z}[X]$. Every ${\Phi }_n$ is irreducible over $\mathbb{Q}$. For prime $p$, ${\Phi }_p(X+1)=((X+1{)}^p-1)/X$ is Eisenstein at $p$, proving the assertion.

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