Let $f$ be a polynomial over $\mathbb{Z}$ or $\mathbb{Q}$. As $\mathbb{Q}$ can be embedded into the field of $p$-adic numbers $\mathbb{Q}_p$ for any prime $p$, the polynomial $f$ can also be considered as an polynomial over $\mathbb{Q}_p$.
Question: Is there a good (sufficient or characterizing) criterion to determine whether $f$ is irreducible over $\mathbb{Q}_p$ or not?
Motivation: In Chapter 10 of these lecture notes on local fields, it turns out that $L=\mathbb{Q}_p(\sqrt{p})$ has ramification degree $e_{L/\mathbb{Q}_p}\geq 2$, so $[L:\mathbb{Q}_p] = 2$. This must mean that the polynomial $f = x^2 – p$ must be irreducible. I can easily conclude the irreducibility over $\mathbb{Q}$ by using Eisenstein's criterion. But I do not know how to make a transition to the larger field $\mathbb{Q}_p$.
Any help is really appreciated.
Best Answer
Eisenstein's criterion works just fine because $\Bbb Z_p$ is an integral domain and $p\Bbb Z_p$ its prime ideal.