Orders of decomposition and inertia groups

Given: The data and hypotheses of the statement.

Decomposition inertia exact sequence: For finite Galois L/K and fixed nonzero $P\mid p$, reduction gives the exact sequence $1⟶I(P\mathrm{/}p)⟶D(P\mathrm{/}p)⟶\mathrm{Gal}(\kappa (P)\mathrm{/}\kappa (p))⟶1.$ In particular $D(P\mathrm{/}p)\mathrm{/}I(P\mathrm{/}p)$ is canonically the residue Galois group.

In a finite Galois extension the Galois group acts transitively on the $g$ primes above $p$, the stabilizer of $P$ is $D(P\mathrm{/}p)$, and orbit-stabilizer gives $\mathrm{\mid }D(P\mathrm{/}p)\mathrm{\mid }=\mathrm{\mid }G\mathrm{\mid }\mathrm{/}g$ (Galois action on primes above a prime is transitive, Orbit-stabiliser: $G\mathrm{/}{G}_x\to G\cdot x$, $g{G}_x\mapsto g\cdot x$, is a well-defined bijection).

For the $g$ primes above $p$, the common ramification index and residue degree satisfy $efg=[L:K]=\mathrm{\mid }G\mathrm{\mid }$ (Galois prime decomposition efg).

A finite extension of a finite field of order $q$ is Galois with cyclic Galois group generated by $x\mapsto x^q$: Let ${\mathbb{F}}_q$ be a finite field of order $q$ and let $E$ be a finite field having ${\mathbb{F}}_q$ as a subfield, with $[E:{\mathbb{F}}_q]=n$ (def-extension-degree-and-finite-extension). Then $E\mathrm{/}{\mathbb{F}}_q$ is a finite Galois extension (def-finite-galois-extension-and-galois-group) and $\mathrm{Gal}(E\mathrm{/}{\mathbb{F}}_q)=⟨{\sigma }_q⟩$ is cyclic of order $n$, generated by the relative Frobenius ${\sigma }_q\text{\hspace{0.17em}⁣}:x\mapsto x^q$ (def-relative-frobenius-of-a-finite-field-extension).