Some days ago I was reading Serre's book on local class field theory and I got stuck on a point he makes in chapter seven after having defined the transfer homormophism.
First I report some notation: Let $A$ be a Dedekind domain, $K$ its field of fractions, $L$ a finite galois extension of $K$, $L^{ab}$ the largest abelian subextension of $L$. Then let $\frak{p}$ be a prime ideal in $A$, unramified in $L$.
Consider a subextension $E/K$ of $L/K$ and call its maximal abelian subextension $E^{ab}$. Also define ${\frak{p}}_{E}$ the ideal generated by $\frak{p}$ in $E$. Furthermore, we say that $H=Gal(L/E)$ and $G=Gal(L/K)$.
By $({\frak{p}},L/K)$ and $({\frak{p}}_{E},E/K)$ we denote the Artin symbols.
Question: Why are the double cosets $Hx_{i}S$, where $S$ denotes the subgroup of $G$ generated by $({\frak{p}},L/K)$, in bijective correspondence with the prime ideals of $E$ which lie over $\frak{p}$?
I have tried to make this correspondence explicit but I have not yet managed.
The only thing I have thought about is that if $E/K$ is normal then its galois group is $G/H$ and hence the double cosets are in bijection with the cosets of the subgroup $S/H$ which, since the prime $\frak{p}$ is unramified, correspond to the primes lying over it in $E$.
I don't know if this is correct and I wouldn't know where to start with a non galois sub extension.
Furthermore Serre claims, since he is triyng to calculate the Ver homormorphism, that $x_{i}.({\frak{p}},L/K)^{f_{i}}.x_{i}^{-1}$ is precisely $({\frak{p}}_{i},E^{ab}/E)$, where ${\frak{p}}_{i}$ is a prime in $E$ lying over ${\frak{p}}$ and $f_{i}$ is the minimum such that $x_{i}.({\frak{p}},L/K)^{f_{i}} \equiv x_{i}$ (Mod $H_{i}$).
I'm afraid that this second point is quite mysterious to me.