Showing $x^n+f(z)\in\mathbb{Q}[x,z]$ is irreducible if $f$ has no repeated factors

Question

There is a polynomial $f(z)\in \mathbb{Q}[z]$ that has no repeated factors and I must thus show that $x^n+f(z)$ is irreducible.

I know that I can use Eisenstein's Criterion somehow but I was a bit confused on how to apply it since this polynomial is in $\mathbb{Q}[x][z]$ which is a polynomial ring over two variables rather than one. Thanks!

Answer 1

Let $p:=g(z) \in \mathbb{Q}[z]$ be an irreducible factor of $f(z)$. Then $p$ is a prime in $\mathbb{Q}[z]$, and $p^2\nmid f(z)$ as $f$ has no repeated roots. Thus, since $R= \mathbb{Q}[z]$ is a UFD, by Eisenstein's Criterion, $x^n+f(z)$ is irreducible in $R[x]$, and then irreducible in $\mathbb{Q}[x,z].$