Classical Galois theory classifies field extensions. It is a special case of a classification of locally constant sheaves in a topos by permutations representations? of the fundamental groupoid/fundamental group.
We discuss the classical/traditional case of Galois theory, which concerns the classification of field extensions. Below in Galois theory for schemes and then in Galois theory in a topos we discuss how this is a special case of a more general concept of Galois theory in a topos.
If is a Galois extension, we define the Galois group to be
This means that we have
We call separable if it has no multiple zero in .
We call separable over if the irreducible polynomial of over is separable.
A subfield is called separable over if each is separable over .
We call normal over if for each the polynomial splits completely in linear factors in .
Let be a field and a subfield of
Denote by the set of subfields of for which is a finite Galois extension of . Then , when partially ordered by inclusion is a directed poset.
The following assertions are equivalent:
is a Galois extension of .
There is a set of separable polynomials such that is the splitting field of over .
If these conditions are satisfied, then there is a group isomorphism
Since each group is finite, the above isomorphism can be used to equip with a profinite topology (i.e. take the limit in the category of topological groups, where each has the discrete topology), making it into a profinite group. We henceforth consider as a profinite group in this way.
More precisely, the maps
are bijective and inverse to each other. This correspondence reverses the inclusion relation: corresponds to and to .
If corresponds to , then we have
This appears for instance as Lenstra, theorem 2.3.
This suggests that more fundamental than the subgroups of a Galois group are its quotients by subgroups, which can be identified with transitive -sets. This naturally raises the question of what corresponds to non-transitive -sets.
A collection of elements of is called a basis of (over ) if for every there is a unique collection of elements of such that for all but finitely many and .
Let be a finitely generated free -module with basis and let be -linear. Then
for certain , and the trace of is defined by
This is an element of that only depends on , and not on the choice of basis. It is easily checked that the map is -linear.
Let be a ring, an -algebra, and suppose that is free with finite rank as an -module. For every the map defined by is -linear, and the trace or is defined to be . The map is easily seen to be -linear and to satisfy for .
If for an -algebra the the morphism is an isomorphism we call separable over , or a free separable -algebra if we wish to stress the condition that is finitely generated and free as an -module.
Recall the notion of separable elements
From xyz it follows that the inclusion is Galois.
The Galois group is called the absolute Galois group of .
for each the open subschemes of is affine,
and equal to , where is a free separable -algebra.
In this situation we also say that is a finite étale covering of .
A morphism from a finite étale covering to a finite étale covering is a morphism of schemes such that .
This defines the category of finite étale covers of .
Let be a connected scheme. Then there exists a profinite group – the fundamental group of – uniquely determined up to isomorphism, such that the category of finite étale coverings is equivalent to the category of finite permutation representations of (finite sets, with the discrete topology, on which acts continuously).
The profinite group, , is often called the étale fundamental group of the connected scheme . In SGA1, Grothendieck also considers coverings with profinite fibres, and a profinitely enriched fundamental groupoid. In the above the actual group depends on the choice of a fibre functor given by a geometric point of . Different choices of fibre functor produce isomorphic groups. Taking two such fibre functors yields a -torsor for either version of . This is important in attacks on Grothendieck's section conjecture.
The disjoint union of copies of corresponds, under this theorem, to a finite set of elements on which acts trivially.
The fact that for there are no other finite étale coverings of is thus expressed by the group being trivial .
The same is true for , where is an algebraically closed field.
In particular, if is a finite field, then .
More generally, if , , then is the Galois group, over , of the maximal algebraic extension of that is unramified at all non-zero prime ideals of not dividing .
We denote by a field. It is our purpose to show that the opposite category of the category of free separable -algebras is equivalent to the category of finite -sets, for a certain profinite group . This is a special case of the main theorem 4, with . In the general proof we shall use the contents of this section only for algebraically closed . In that case, which is much simpler, the group is trivial, so that the category of finite -sets is just the category of finite sets.
In SGA1, Grothendieck introduced an abstract formulation of the above theory in terms of Galois categories. A Galois category is a category, , satisfying a small number of properties together with a fibre functor , preserving those properties. The theory is more fully described in the entry on Grothendieck's Galois theory.
The étale morphisms corresponds precisely to the locally constant sheaves on with respect to the etale topology, in that it is equivalently a morphism for which there is an etale cover such that is a constant sheaf on each .
the sheaf topos over it. This topos is a
The Galois group is the fundamental group of the topos.
In the context of higher topos theory, there are accordingly higher analogs of Galois theory.
For aspects see
Between January and June 1981, Grothendieck wrote about 1600 manuscript pages of a work with the above title. The subject is the absolute Galois group, of the rational numbers and its geometric action on moduli spaces of Riemann surfaces. This is discussed in Long March?. Other entries that relate to this include anabelian geometry, children's drawings (in other words Dessins d’enfants, which is the study of graphs embedded on surfaces, their classification and the link between this and Riemann surfaces) and the Grothendieck-Teichmuller group.
The anabelian question is: how much information about the isomorphism class of an algebraic variety, is contained in the étale fundamental group of ? Grothendieck calls varieties which are completely determined by their étale fundamental group, anabelian varieties. His anabelian dream was to classify the anabelian varieties in all dimensions over all fields. This can be seen to relate to questions of the étale homotopy types of varieties.
Tim: I have a feeling that this anabelian question should have a form that generalises to higher dimensions. (Not that I can shed much light on progress in dimension one.) Perhaps there is an anabelian version of the homotopy hypothesis or something of that nature.
Lascar group (a Galois group of first order theories)
Lecture notes on the Galois theory for schemes are in
Some of the material above is taken from this.
A comprehensive textbook is
The locally simply connected case is discussed for instance in
Tamás Szamuely, Galois groups and fundamental groups, Cambridge Studies in Adv. Math.