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.


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