# Double cosets (Neukirch's Algebraic Number Theory)

This is a question from Neukirch's Algebraic Number Theory, Ch.1 $$\S$$9.

Let $$A$$ be a Dedekind domain with quotient field $$K$$, $$L$$ a finite separable field extension of $$K$$ and $$B$$ the integral closure of $$A$$ in $$L$$. Let $$N/K$$ be a Galois closure of $$L/K$$ and set $$G=\operatorname{Gal}(N/K)$$, $$H=\operatorname{Gal}(N/L)$$.

Let $$\mathfrak p$$ be a prime ideal of $$A$$ and $$P_\mathfrak p$$ the set of prime ideals of $$L$$ above $$\mathfrak p$$. If $$\mathfrak P$$ is a prime ideal of $$N$$ above $$\mathfrak p$$ let $$G_\mathfrak P$$ be the decomposition subgroup of $$\mathfrak P$$ over $$K$$.

Consider the map $$H\backslash G/G_\mathfrak P\rightarrow P_\mathfrak p$$ defined by $$H\sigma G_\mathfrak P \mapsto \sigma \mathfrak P\cap L$$.

Question: how do I show that it is bijective?

I can show that it is well-defined and injective but surjection seems more tricky. I know that the number of distinct primes in $$N$$ above $$\mathfrak p$$ is $$[G:G_\mathfrak P]$$. Does this imply that the two sets are of equal size?

Many thanks.

Surjectivity is easy, as clearly when $$\sigma$$ traverses all the cosets of the decomposition group of $$G_P$$, $$G_P\cap L$$ traverses all prime ideals above $$p$$ in $$L$$, as each such prime has some prime $$P$$ in $$N$$ above it.
Injectivity follows from the fact that if $$P$$ and $$Q$$ are two different primes in $$N$$ above $$p$$ having the same intersection with $$L$$, then if $$\sigma(P) = Q$$, we can show that $$\sigma$$ is in the same doubled coset as the identity. Proof is as follows: Denote by $$q$$ the intersection of $$P$$ with $$L$$ ($$P\cap L = Q\cap L = q$$), then since $$N|L$$ is Galois, and $$P,Q$$ are above $$q$$, there is an element of $$H$$, which stabilizes $$L$$ and takes $$P$$ to $$Q$$, let us denote it $$\tau$$. Then $$\tau^{-1}\sigma(P) = P$$, implying that $$\tau^{-1}\sigma\in G_P$$ (apologies for not knowing how to write Gothic $$P$$). We therefore have: $$\sigma\in \tau G_P \subseteq HG_P$$, which is the doubled coset of the identity.