Exercise 21, § 10

Let $f(X) = X^n + a_1 X^{n-1} + \ldots + a_n$ be a polynomial of Z[X] and p a prime number. Suppose that $a_i \equiv 0 \pmod{p}$, $1 \leq i \leq n$ and that $a_n \not\equiv 0 \pmod{p^2}$. Show that $f(X)$ is irreducible in Q[X]. (Let $f(X) = P(X) Q(X)$ be a decomposition in Q[X]. Show that if P and Q are monic, they have integer coefficients. Reduce modp.)