degree extension over filed of $p$-adic numbers Let $K = \mathbb{Q}(\theta)$ be a numberfield and $[K:\mathbb{Q}]=n$. When $\mathbb{Q}_p$ is the field of  $p$-adic numbers and $K_p=\mathbb{Q}_p(\theta)$, what about $[K_p : \mathbb{Q}_p]$?

I think an illustrative example is to look at cyclotomic extensions. Hensel's Lemma states that if a primitive polynomial $f$ in $\mathbb{Z}_p[x]$ admits a modulo $p$ factorization
$$f \equiv \bar g \bar h \mod p$$
Then it admits a factorization
$$f = gh$$
Such that $\deg g = \deg \bar g$ and $g \equiv \bar g$ and $h \equiv \bar h$.
Now we know that our residue field is $\mathbb{F}_p$ and so $x^{p-1}-1$ splits and by Hensel's Lemma it must split in $\mathbb{Z}_p$ so we already contain these roots of unity. 
Now what does this have to do with your question. We have $[\mathbb{Q}(\zeta_p) : \mathbb{Q}] = p$ but $[\mathbb{Q}_p(\zeta_p) : \mathbb{Q}_p] = 1$. This is not always true (check other roots of unity) but does give an example saying that it's not an immediate or easy relationship.
