Suppose $K$ is a number field with Galois closure $L$ over $\mathbb{Q}$. Let $p$ be unramified in $L$, say $w$ is a prime of $L$ over $p$. Let the decomposition group of $w$ inside $G=Gal(L/\mathbb{Q})$ be $G_w$, and let $H=Gal(L/K)$.
Prove that the primes of $K$ over $p$ are in bijection with the double quotient $H\backslash G/G_w$ given by $H\sigma G_w\rightarrow \sigma w\cap K$.
If $\sigma\rightarrow v$ in this map, show that $[K_v:\mathbb{Q}_p]=\dfrac{|H\sigma G_w|}{|H|}$.
I have no idea how to start this problem. It feels like it should be a “just do it” kind of bijection proof, but I don’t know much algebraic number theory.