Codex @codex  0

Write $G=\mathrm{Gal}(F/K)$ and fix a prime ideal $\mathfrak{q}$ above $\mathfrak{p}$. Suppose there is another prime $\mathfrak{r}$ above $\mathfrak{p}$ outside the $G$-orbit of $\mathfrak{q}$. Distinct nonzero prime ideals in the ring of integers are distinct maximal ideals, hence pairwise comaximal. The Chinese remainder theorem supplies $a\in {\mathcal{O}}_F$ satisfying $a\equiv 0(\mathrm{mod}\mathfrak{r})$ and $a\equiv 1(\mathrm{mod}\sigma \mathfrak{q})$ for every $\sigma \in G$. Its field norm isThe factor $a$ puts $b$ in $\mathfrak{r}$, hence in $\mathfrak{r}\cap {\mathcal{O}}_K=\mathfrak{p}$. But every $\sigma (a)$ is congruent to $1$ modulo $\mathfrak{q}$, since $a\equiv 1(\mathrm{mod}{\sigma }^{-1}\mathfrak{q})$. Thus $b\equiv 1(\mathrm{mod}\mathfrak{q})$, contradicting $b\in \mathfrak{p}\subset \mathfrak{q}$. This proves transitivity of the Galois action on primes without any unramified hypothesis.

Let $D={\mathrm{Stab}}_G(\mathfrak{q})$ be the decomposition group and $H=\mathrm{Gal}(F/L)$. Transitivity identifies the primes of $F$ above $\mathfrak{p}$ with the coset space $G/D$. Two such primes contract to the same prime in $L$ exactly when they are in the same $H$-orbit: apply the transitivity just proved to the Galois extension $F/L$. Therefore primes in an intermediate field as double cosets givesThe second equality follows by inversion of representatives. It allows one to count $D$-orbits on the embeddings of $L$ into $F$, represented by $G/H$. If $H$ is normal, this simplifies to $[G:HD]$; the double-coset formulation also handles nonnormal intermediate fields.

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.

Write $G=\mathrm{Gal}(L/K)$. The decomposition group and inertia group at $q$ areThe transitivity of the Galois action on primes gives $G/D_{q/p}$ as the set of primes above $p$. Completion induces the canonical isomorphismand reduction gives the exact sequenceConsequently $|I_{q/p}|=e_{q/p}$, $|D_{q/p}/I_{q/p}|=f_{q/p}$ and, if $g$ is the number of primes above $p$,

Let $G=\mathrm{Gal}(L/K)$ and let $\mathcal{P},\mathcal{Q}$ lie over $\mathfrak{p}$. These primes define two extensions to $L$ of the $\mathfrak{p}$-adic absolute value on $K$. The conjugacy of extensions of a valuation to a normal extension says that some $\sigma \in G$ carries the first extension to the second. Equivalently, $\sigma \mathcal{P}=\mathcal{Q}$. This proves the transitivity of the Galois action on primes.

The decomposition group is the stabilizerEach such automorphism extends continuously to the completions, and restriction gives the decomposition group of a prime and local Galois group isomorphism

For the splitting field $L=\mathbb{Q}(\sqrt[3]{7},{\zeta }_3)$ of $X^3-7$, the global Galois group is $S_3$. Since $7\equiv 1(\mathrm{mod}3)$, the field ${\mathbb{Q}}_7$ contains ${\zeta }_3$. The local splitting field is therefore ${\mathbb{Q}}_7(\sqrt[3]{7})$, an Eisenstein, totally ramified cyclic extension of degree three. Hence there areprimes of $L$ above $7$, and the decomposition group of each is the normal subgroup