Bound on prime and ramification index in local fields.

Suppose $K/\mathbb{Q}$ is a finite extension (degree $n$ say). Choose a prime $\Lambda$ of $K$ lying above $p$ and suppose $K_{\Lambda}/\mathbb{Q}_p$ has ramification index $e$.

How "likely" is it that $p>e+1$ holds?

Eventually I am going to want to choose $p$ to satisfy this condition as well as not lying in certain arithmetic progressions.

• You mean beyond the fact that $e\le n$, so that all but finitely many primes $p$ have that property? – Lubin Feb 27 '15 at 23:39
• Oh of course...how stupid of me! I forgot the tight link between global and local. Thanks! – fretty Mar 1 '15 at 8:23