How to show this polynomial is irreducible?

Question

In this question, it is mentioned that the polynomial $x^p - x - 1/p$ is irreducible over $\mathbb{Q}_p[x]$, but I do not see why this should be true. Would someone be able to provide a proof of this claim?

Answer 1

Let $P(x)=x^p-x-\frac1p$. If $P(x)$ is reducible in $\mathbb{Q}[x]$, then so is the polynomial$$Q(x)=-px^pP\left(\frac1x\right)=x^p+px^{p-1}-p.$$But that's not the case, by Eisenstein's criterion.