A extending the p-adic valuation to a quadratic extension of $\mathbb{Q}_p$ I'm trying to solve the following problem. Prove that, if $d \in \mathbb{Z}_p$ is non-square, then
$|a + b \sqrt{d}|p = |a^2 − b^2d|^{1/2}_p$ , for any $a, b \in \mathbb{Q}p$,
defines a non-Archimedean valuation on $\mathbb{Q}_p(\sqrt{d})$ which extends the usual $|. |_p$ on $\mathbb{Q}_p$.
The hint we are given is to show that $|\alpha|_p\le 1 \Rightarrow |\alpha +1|_p \le 1$. I've managed to reduce this to different cases, and the only case where I'm still having trouble $|a|_p=|b|_p$. Apparently I need to use Hensel's Lemma somehow, but I don't know how. Can anyone give me a hint? 

Sure, here’s a hint. Use the good form of Hensel to show this: let $\alpha$ be algebraic over $\Bbb Q_p$, with minimal polynomial $f(X)$ whose constant term is $c$. If $c\in\Bbb Z_p$ (the $p$-adic integers), then all the coefficients of $f$ are in $\Bbb Z_p$. You definitely do not need to divide the problem into cases.
