I have the following problem: suppose $F/K$ is an abelian (Galois) extension of number fields, Galois group G, and $\mathfrak{p}$ is a prime of K, $\mathfrak{P}$ a prime of F dividing $\mathfrak{p}$; then let M be the subfield of F fixed by the decomposition group $G_{\mathfrak{P}} \subset G$. Then I want to show that $\mathfrak{p}$ splits completely in M; all ramification indices $e_i$ and residue class degrees $f_i$ are equal to 1 in M/K.
So, I know that the degree of an extension is the product of e, f and the number of primes (with all $e_i$ are equal, say to e, all $f_i$ equal to f). I know $|G_{\mathfrak{P}}|=ef$ is the product of the ramification index and residue class degree. However, I'm struggling to show that $\mathfrak{p}$ splits completely in M. I feel like I have a lot of options and I'm not sure what's best: I could try to show directly that each $e_i$ and $f_i=1$ , or try to play around with the tower law and the degree $[M:K]$, or try to make use of the properties of the decomposition group or something similar. Nothing I've done so far has got me anywhere however - could anyone help please? Thanks in advance! -Tom
Edit: I was an idiot and accidentally cleared my cookies without thinking, sorry. Have made an account to stop this happening again - could you possibly link this to this account so I can add comments? Apologies!