Decomposition and inertia in towers

Given: The data and hypotheses of the statement.

Conjugacy of decomposition and inertia groups: In finite Galois L/K, if $\sigma P=P^{\mathrm{\prime }}$ above a nonzero p, then $D(P^{\mathrm{\prime }}\mathrm{/}p)=\sigma D(P\mathrm{/}p){\sigma }^{-1},I(P^{\mathrm{\prime }}\mathrm{/}p)=\sigma I(P\mathrm{/}p){\sigma }^{-1}\mathrm{.}$ The residue actions correspond under $\kappa (P)\to \kappa (P^{\mathrm{\prime }})$, $\bar{a}\mapsto \overline{\sigma a}$.

Galois action on primes above a prime is transitive: Let L/K be a finite Galois extension of number fields and p a nonzero prime of ${\mathcal{O}}_K$. Then $G=\mathrm{Gal}(L\mathrm{/}K)$ acts transitively on the primes P above p.

Orders of decomposition and inertia groups: For finite Galois L/K and nonzero $P\mid p$, writing e and f for its ramification index and residue degree, $\mathrm{\mid }D(P\mathrm{/}p)\mathrm{\mid }=ef,\mathrm{\mid }I(P\mathrm{/}p)\mathrm{\mid }=e,\mathrm{\mid }D(P\mathrm{/}p)\mathrm{/}I(P\mathrm{/}p)\mathrm{\mid }=f\mathrm{.}$ The prime P is unramified over p if and only if its inertia group is trivial.

Ramification and residue degrees in towers: For $M\mathrm{/}L\mathrm{/}K$ and $\mathfrak{Q}\mid \mathfrak{P}\mid \mathfrak{p}$, $e(\mathfrak{Q}\mathrm{/}\mathfrak{p})=e(\mathfrak{Q}\mathrm{/}\mathfrak{P})e(\mathfrak{P}\mathrm{/}\mathfrak{p}),f(\mathfrak{Q}\mathrm{/}\mathfrak{p})=f(\mathfrak{Q}\mathrm{/}\mathfrak{P})f(\mathfrak{P}\mathrm{/}\mathfrak{p})\mathrm{.}$

Normal subgroups, conjugate fields, and quotient groups in the Galois correspondence: Let $K\mathrm{/}F$ be finite Galois, let $G=\mathrm{Gal}(K\mathrm{/}F)$, let $H\le G$, and put $E=K^H$. For every $\sigma \in G$, $\mathrm{Gal}(K\mathrm{/}\sigma (E))=\sigma H{\sigma }^{-1}\mathrm{.}$ An intermediate field $E\mathrm{/}F$ is Galois exactly when its corresponding subgroup is normal. In that case restriction gives a surjective homomorphism $G\to \mathrm{Gal}(E\mathrm{/}F)$ with kernel $H$, and hence $\mathrm{Gal}(E\mathrm{/}F)\cong G\mathrm{/}H\mathrm{.}$