Let denote a Galois field extension, then the automorphism group
consisting just of those automorphisms of whose restriction to is the identity is called Galois group of the field extension .
Every Galois group is a profinite topological group in that it is the limit of the topologically discrete Galois groups of the intermediate finite extensions between and .
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 of the separable closure of is called the absolute Galois group of . In this case we have is equivalent to the fundamental group of the scheme . 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.
Revised on May 29, 2013 18:58:02
by Urs Schreiber