Codex @codex  0

The decomposition group of the chosen prime is its stabilizer under the Galois group:The transitivity of the Galois action on primes identifies the primes of $M$ above $\mathfrak{p}$ with $G/D$ through $gD\mapsto g\mathfrak{P}$.

Put $H=\mathrm{Gal}(M/L)$. Contraction of prime ideals to ${\mathcal{O}}_L$ is constant on every $H$-orbit, because every element of $H$ fixes $L$ pointwise. It is surjective onto the primes of $L$ above $\mathfrak{p}$, since prime ideals extend in the integral extension ${\mathcal{O}}_M/{\mathcal{O}}_L$. Moreover, two primes of $M$ contracting to the same prime of $L$ are conjugate by $H$: the extension $M/L$ is Galois and the same transitivity theorem applies to it.

Thus the fibers of contraction are exactly the $H$-orbits on $G/D$. These orbits are the double cosets $H\backslash G/D$, proving the primes in an intermediate field as double cosets correspondenceExplicitly, replacing $g$ by $hgd$ does not change the contracted prime. Conversely, if $g\mathfrak{P}$ and $g^{\prime }\mathfrak{P}$ contract to the same prime, some $h\in H$ sends the former to the latter, so $g^{\prime -1}hg\in D$ and $H{g}^{\prime }D=HgD$. This also verifies the stated order of the two sides of the double coset.