Past exam of the mathematics course of the University of Cambridge / 2004 / ib / Paper 4 / 2F / Solution

A polynomial in $\mathbb{Z}[x]$ is a primitive polynomial when the greatest common divisor of its coefficients is one. The Gauss lemma for polynomials states that a product of primitive integer polynomials is primitive, and that a primitive positive-degree integer polynomial is reducible over $\mathbb{Q}$ exactly when it factors into positive-degree integer polynomials. More generally, polynomial content is multiplicative up to sign.

The Eisenstein criterion states that a primitive polynomial $a_n{x}^n+\cdots +a_0\in \mathbb{Z}[x]$ is irreducible over $\mathbb{Q}$ if there is a prime $p$ such that $p\nmid a_n$, $p\mid a_j$ for every $j<n$, and $p^2\nmid a_0$. These criteria permit integer coefficient tests for rational irreducibility; the following parts apply them with their exact divisibility hypotheses.