Galois group (reviewed)

Let $K$ be a finite degree $n$ separable extension of a field $F$, and $K^{gal}$ be its Galois (or normal) closure. The Galois group for $K/F$ is the automorphism group $\Aut(K^{gal}/F)$.

This automorphism group acts on the $n$ embeddings $K\hookrightarrow K^{gal}$ via composition. As a result, we get an injection $\Aut(K^{gal}/F)\hookrightarrow S_n$, which is well-defined up to the labelling of the $n$ embeddings, which corresponds to being well-defined up to conjugation in $S_n$.

We use the notation $\Gal(K/F)$ for $\Aut(K/F)$ when $K=K^{gal}$.

There is a naming convention for Galois groups up to degree $47$.