Finite Galois extension

A finite extension $L/K$ is Galois when it has $[L:K]$ automorphisms fixing $K$.

For a finite Galois extension $L/K$ with group $G$, there is $\theta \in L$ such that the elements $g(\theta )$ form a $K$-basis. Equivalently the additive $K[G]$-module $L$ is a regular representation. Over p-adic fields, its p-adic lattices are thus commensurable with regular lattices; this is useful in Herbrand quotient calculations after applying a p-adic logarithm to deep principal units.

The Galois group $\mathrm{Gal}(L/K)$ consists of the field automorphisms of $L$ that fix $K$ pointwise. For a finite Galois extension, its order is $[L:K]$.

A separable extension has $[L:K]$ embeddings in an algebraic closure, and normality makes every one an automorphism of $L$. Hence a finite normal separable extension is Galois.

An abelian extension is a Galois extension whose Galois group is an abelian group.