Galois extension and Galois group

A finite field extension $K/k$ is Galois if it is normal and separable. Its Galois group is

\operatorname{Gal}(K/k)=\operatorname{Aut}_k(K),

and the fixed field of the whole group is $k$ .

Normality means that every $k$ -embedding $K\hookrightarrow\overline k$ has image $K$ . Separability means that every element has a separable minimal polynomial over $k$ .

Infinite extensions

An algebraic extension can be Galois without being finite. Its Galois group has the Krull topology

\operatorname{Gal}(K/k) \simeq \varprojlim_{L/k\ \mathrm{finite\ Galois}} \operatorname{Gal}(L/k),

and is profinite. The absolute Galois group is $\Gamma_k=\operatorname{Gal}(k_s/k)$ .

Continuous finite quotients of $\Gamma_k$ correspond to finite Galois extensions. Continuous $\ell$ -adic representations can have infinite image and therefore do not generally factor through one finite extension.

Langlands role

Finite Galois groups act on split root data in descent constructions. Weil groups and absolute Galois groups supply the parameter side of local and global Langlands. The $L$ -group usually uses a Weil group whose action on the pinned dual group factors through a finite Galois quotient.

References

Jean-Pierre Serre, Galois Cohomology, Springer, 1997.