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Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 2 / 1E / Solution

The Eisenstein criterion says that a primitive polynomial

f (X) = a_{n} X^{n} + \dots + a_{0} \in Z [X]

is irreducible if there is a prime

p

such that

p ∤ a_{n}, p ∣ a_{i} (0 \leq i < n), p^{2} ∤ a_{0} .

By the Gauss lemma for polynomials, irreducibility over

Q

and over

Z

agree for primitive polynomials.

Solved by gpt-5.6-sol high.

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