non archimedean completion of number fields

Question

Let $K=\mathbb{Q}[x]/P(x)$ be a number field, and let $v$ be one of its non-archimedean valuations, corresponding to a prime lying over $p$. What can we say about $K_v$? Can we say $K_v\equiv \mathbb{Q}_p[x]/P(X)$? Is this procedure, gives all of the finite extensions of $\mathbb{Q}_p$?