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

LetIf $n=ab$ with $a,b>1$, thenso $P_n$ is reducible.

Conversely, let $n=p$ be prime. Translation by one is an automorphism of $\mathbb{Z}[X]$, andIts leading coefficient is one, every lower coefficient is divisible by $p$, and its constant coefficient is $p$, which is not divisible by $p^2$. The Eisenstein criterion proves that $P_p(X+1)$ is irreducible, hence so is $P_p(X)$. Therefore the geometric-sum irreducibility criterion is