Counterexample to Eisenstein criterion

Question

We know Eisenstein criterion about irreducibility of polynomials: if $q(x) = x^n + a_{n-1}x^{n-1} + \dots +a_0 \in \mathbb{Z}[x]$ is such that $\exists p$ prime number with $ p \mid a_{i} \ \forall i \ , \ p^2 \nmid a_0$ then $q(x)$ is irreducible over $\mathbb{Q}$.
The question is : how to find a counterexample in the case $p$ is not a prime ?

Answer 1

You mean like $(x-2)(x+2)=x^2-4$ for $p=4$?

Method used to find this counterexample:

Thought: Keep things simple. Just have a monic leading term and then a constant. Asked self: Do I know any easy factorizations for binomials? Thought: Of course: $(a-b)(a+b)=a^2-b^2$ Completed: Use $a=x$ and pick any $b$ but make sure $b^2\neq 1$ and use $p=b^2$