Why order of decomposition group＝$[L:K]/$(number of prime ideal above $ \mathfrak p$)

Question

Let $L/K$ be a finite galois extension of number field.

Let $\mathfrak p$ be prime ideal of $K$ .$$D_{\mathfrak p} = \{ \sigma \in G : \sigma(\mathfrak p) = \mathfrak p\}$$ be decomposition group.

Could you tell me proof of $♯D_{\mathfrak p}＝[L:K]/$(number of prime ideal above $ \mathfrak p$) holds ?

Maybe simple observation may deduce this, I'm having trouble relating decomposition group and number of prime ideal above $ \mathfrak p$.

Answer 1

It's well-known that the action of $G:=\mathrm{Gal}(L/K)$ on the set $X$ of prime ideals above $\mathfrak{p}$ is transitive. Now fix a prime ideal $\mathfrak{P}$ above $\mathfrak{p}$. Then $D_\mathfrak{P}$ is the stabiliser of $\mathfrak{P} \in X$ and so the orbit-stabiliser theorem implies what we want.