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Galois group

Definition

Definition

Let $K\hookrightarrow L$ denote a Galois field extension?, then the automorphism group

$$\mathrm{Gal}(K\hookrightarrow L):={\mathrm{Aut}}_K(L)$$Gal(K\hookrightarrow L):=Aut_K(L)

consisting just of those automorphisms of $L$ whose restriction to $K$ is the identity is called Galois group of the field extension $K\hookrightarrow L$.

Every Galois group $\mathrm{Gal}(K\hookrightarrow L)={\lim }_{K\hookrightarrow E\hookrightarrow L}\mathrm{Gal}(K\hookrightarrow E)$ is a profinite topological group in that it is the limit of the topologically discrete Galois groups of the intermediate finite extensions between $K$ and $L$.

The just defined Galois group is the one occurring in the classical Galois theory for fields. The analog of the Galois group in Galois theory for schemes is a fundamental group (of a scheme) and is rarely called a ”Galois group”.

The Galois group $\mathrm{Gal}(K\hookrightarrow K_s)$ of the separable closure of $K$ is called the absolute Galois group of $K$. In this case we have $\mathrm{Gal}(K\hookrightarrow K_s)\simeq {\pi }_1(\mathrm{Spec}K)$ is equivalent to the fundamental group of the scheme $\mathrm{Spec}K$. In particular the notion of fundamental group (of a point of) a topos generalizes that of Galois group. This observation is the starting point and motivating example of Grothendieck's Galois theory and more generally of that of homotopy groups in an (infinity,1)-topos.

If the scheme, moreover, is a group scheme (i.e. endowed with a group structure) modules over the Galois group, which are called Galois modules, play an important role in algebraic number theory.

category: Galois theory