Show that $x^n + ax + p$ is irreducible, $p$ prime, $a \in \mathbb Z, \lvert a \rvert < p - 1$

Question

Show that the polynomial $x^n + ax + p$ is irreducible, $p$ prime, $a \in \mathbb Z, \lvert a \rvert < p - 1$.
I tried reduction, Eisenstein and linear transformation (plugging in $x+1$, $x+p$) but nothing did work. Any solutions?

Answer 1

If $x^n+ax+p = fg$ were reducible, with nonconstant monic $f,g\in \mathbb{Z}[x]$. Since $p$ is prime, we can assume the constant term of $f$ be $\pm 1$. So products of roots of $f$ has absolute value $1$, one of the root $r$ must satisfy $|r|\leq1$.

However, $|r^n + ar| \leq 1+|a| < p$, contradicting to the fact that $r^n+ar+p=0$.