Grothendieck's Galois theory
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In mathematics, Grothendieck's Galois theory is a highly abstract approach to the Galois theory of fields, developed around 1960 to provide a way to study the fundamental group of algebraic topology in the setting of algebraic geometry. It provides, in the classical setting of field theory, with an alternative perspective to that of Emil Artin, whose treatment from about 1930 became standard.
The approach of Alexander Grothendieck is concerned with the category theory properties that characterise the categories of finite G-sets for G a fixed profinite group. For example G might be the group denoted Z^, which is the inverse limit of the cyclic additive groups Z/nZ - or equivalently the completion of the infinite cyclic group Z for the topology of subgroups of finite index. A finite G-set is then a finite set X on which G acts through a quotient finite cyclic group; so that it is specified by giving some permutation of X.
The connection with Galois groups can be seen when it is realised that Z^ is the pro-finite Galois group of the algebraic closure F* of any finite field F, over F. That is, the automorphisms of F* fixing F are described by the inverse limit, as we take larger and larger finite splitting fields over F. The connection with geometry can be seen when we look at covering spaces of the unit disk with 0 removed: the finite covering realised by the zn map of the disk, thought of by means of a complex number variable z, corresponds to the subgroup n.Z of the fundamental group of the punctured disk.
The theory of Grothendieck, published in SGA1, shows how to reconstruct the category of G-sets from a fibre functor Φ, which in the geometric setting takes the fibre of a covering above a fixed base point (as a set). In fact there is an isomorphism proved of the type G = Aut(Φ), the latter being the group of automorphisms (self-natural equivalences) of Φ. An abstract classification of categories with a functor to the category of sets is given, by means of which one can recognise categories of G-sets for G profinite.
To see how this applies to the case of fields, one has to study the tensor product of fields. Later developments in topos theory make this all part of a theory of atomic toposes.