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.
Even more generally one can define a Galois group associated to a presentable symmetric monoidal stable (infinity,1)-category. There is an analogue of the Galois correspondence in this setting, see Mathew 14.
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.
We call a field extension $K \subset L$ a Galois extension if $K \subset L$ is algebraic and there exists a subgroup $G \subset Aut(L)$ of the automorphism group such that $K \simeq L^G$ is the field of elements that are invariant under $G$.
If $K \subset L$ is a Galois extension, we define the Galois group to be
This means that we have
Let $\bar K$ be a fixed algebraic closure of $K$. If $F \subset K[X] - \{0\}$ is any collection of non-zero polynomials, the splitting field of $F$ over $K$ is the subfield of $\bar K$ generated by $K$ and the zeros of the polynomials in $F$.
We call $f \in K[X]- \{0\}$ separable if it has no multiple zero in $\bar K$.
We call $\alpha \in \bar K$ separable over $K$ if the irreducible polynomial $f^\alpha_K$ of $\alpha$ over $K$ is separable.
A subfield $L \subset K$ is called separable over $K$ if each $\alpha \in L$ is separable over $K$.
We call $L$ normal over $K$ if for each $\alpha \in L$ the polynomial $f^\alpha_K$ splits completely in linear factors in $L[X]$.
Let $K$ be a field and $L$ a subfield of $\bar K$
Denote by $I$ the set of subfields $E$ of $L$ for which $E$ is a finite Galois extension of $K$. Then $I$, when partially ordered by inclusion is a directed poset.
The following assertions are equivalent:
$L$ is a Galois extension of $K$.
There is a set $F \subset K[X] - \{0\}$ of separable polynomials such that $L$ is the splitting field of $F$ over $K$.
$\coprod_{E \in I} E \simeq L$
If these conditions are satisfied, then there is a group isomorphism
where on the right we have the limit over the poset of subfield of the contravariant functor $E \mapsto Gal(E/K)$.
Since each group $Gal(E/K)$ is finite, the above isomorphism can be used to equip $Gal(L/K)$ with a profinite topology (i.e. take the limit in the category of topological groups, where each $Gal(E/K)$ has the discrete topology), making it into a profinite group. We henceforth consider $Gal(L/K)$ as a profinite group in this way.
(main theorem of classical Galois theory)
Let $K \subset L$ be a Galois extension of fields with Galois group $G$. Then the intermediate fields of $K \subset L$ correspond bijectively to the closed subgroups of $G$.
More precisely, the maps
defined by
and
are bijective and inverse to each other. This correspondence reverses the inclusion relation: $K$ corresponds to $G$ and $L$ to $\{id_L\}$.
If $E$ corresponds to $H$, then we have
$K \subset E$ is finite precisely if $H$ is open (in the profinite topology on $G$)
$[E:K] \simeq index[G:K]$ if $H$ is open;
$E \subset L$ is Galois with $Gal(L/E) \simeq H$ (as topological groups);
for every $\sigma \in G$ we have that $\sigma[E]$ corresponds to $\sigma H \sigma^{-1}$;
$L \subset E$ is Galois precisely if $H$ is a normal subgroup of $G$;
$Gal(E/K) \simeq G/H$ (as topological groups) if $K \subset E$ is Galois.
This appears for instance as Lenstra, theorem 2.3.
This suggests that more fundamental than the subgroups of a Galois group $G$ are its quotients by subgroups, which can be identified with transitive $G$-sets. This naturally raises the question of what corresponds to non-transitive $G$-sets.
Let $A$ be a commutative ring and $N$ a module over $A$.
A collection of elements $(w_i)_{i \in I}$ of $N$ is called a basis of $N$ (over $A$) if for every $x \in N$ there is a unique collection $(a_i)_{i \in I}$ of elements of $A$ such that $a_i = 0$ for all but finitely many $i \in I$ and $x = \sum_{i \in I} a_i w_i$.
If $N$ has a basis it is called free (over $A$). If $N$ is free with basis a finite set of cardinality $n$, then we say that $N$ is free with rank $n$ (over $A$). In this case, $N$ is a finitely generated free module.
Let $N$ be a finitely generated free $A$-module with basis $w_1, w_2, \cdots, w_n$ and let $f\colon N \to N$ be $A$-linear. Then
for certain $a_{i j} \in A$, and the trace $Tr(f)$ of $f$ is defined by
This is an element of $A$ that only depends on $f$, and not on the choice of basis. It is easily checked that the map $Tr : Hom_A(N,N) \to A$ is $A$-linear.
Let $A$ be a ring, $B$ an $A$-algebra, and suppose that $B$ is free with finite rank $n$ as an $A$-module. For every $b \in B$ the map $mult_b\colon B \to B$ defined by $mult_b\colon x \mapsto b x$ is $A$-linear, and the trace $Tr(b)$ or $Tr_{B/A}(b)$ is defined to be $Tr(mult_b)$. The map $Tr\colon B \to A$ is easily seen to be $A$-linear and to satisfy $Tr(a) = n a$ for $a \in A$.
The $A$-module $Hom_A(B,A)$ (underlying which is the hom-set in the category of modules) is clearly free over $A$ with the same rank as $B$. Define the $A$-linear map $\phi\colon B \to Hom_A(B,A)$ by
for $x, y \in B$.
If for an $A$-algebra $B$ the the morphism $\phi$ is an isomorphism we call $B$ separable over $A$, or a free separable $A$-algebra if we wish to stress the condition that $B$ is finitely generated and free as an $A$-module.
Recall the notion of separable elements
Let $K$ be a field and $\bar K$ an algebraic closure of $K$. The separable closure $K_S$ of $K$ is defined by
We have that $K_S$ is a subfield of $\bar K$ and that $K_S \simeq \bar K$ precisely if $K$ is a perfect field, in particular if the characteristic of $K$ is 0.
From xyz it follows that the inclusion $K \subset K_S$ is Galois.
The Galois group $Gal(K_S/K)$ is called the absolute Galois group of $K$.
Let $K$ be a field and $\pi_1(Spec K)$ its absolute Galois group. Then there is an equivalence of categories
The classical Galois theory for fields is a special case of a general geometric/topos theoretic statement about locally constant sheaves and the action of the fundamental group on their fibers.
A morphism $f : Y \to X$ of schemes is a finite étale morphism if there exists a covering of $X$ by affine open subsets $U_i = Spec A_i$, such that
for each $i$ the open subschemes $f^{-1}(U_i)$ of $Y$ is affine,
and equal to $Spec B_i$, where $B_i$ is a free separable $A_i$-algebra.
In this situation we also say that $f : Y \to X$ is a finite étale covering of $X$.
A morphism from a finite étale covering $f : Y \to X$ to a finite étale covering $g : Z \to X$ is a morphism of schemes $h : Y \to Z$ such that $f = g \circ h$.
This defines the category $FEt_X$ of finite étale covers of $X$.
Let $X$ be a connected scheme. Then there exists a profinite group $\pi_1(X)$ – the fundamental group of $X$ – uniquely determined up to isomorphism, such that the category of finite étale coverings $FEex$ is equivalent to the category $Fin \pi_1(X) Set$ of finite permutation representations of $\pi_1(X)$ (finite sets, with the discrete topology, on which $\pi_1(X)$ acts continuously).
This appears for instance as Lenstra, main theorem 1.11. It is fully discussed in SGA1.
The profinite group, $\pi_1(X)$, is often called the étale fundamental group of the connected scheme $X$. In SGA1, Grothendieck also considers coverings with profinite fibres, and a profinitely enriched fundamental groupoid. In the above the actual group $\pi_1(X)$ depends on the choice of a fibre functor given by a geometric point of $X$. Different choices of fibre functor produce isomorphic groups. Taking two such fibre functors yields a $\pi_1(X)$-torsor for either version of $\pi_1(X)$. This is important in attacks on Grothendieck's section conjecture.
The disjoint union of $n$ copies of $X$ corresponds, under this theorem, to a finite set of $n$ elements on which $\pi_1(X)$ acts trivially.
The fact that for $X = Spec \mathbb{Z}$ there are no other finite étale coverings of $X$ is thus expressed by the group $\pi_1(Spec \mathbb{Z})$ being trivial .
The same is true for $\pi_1(Spec K)$, where $K$ is an algebraically closed field.
If $K$ is an arbitrary field, then $\pi_1(Spec K)$ is the absolute Galois group of $K$; i.e. the Galois group of the separable closure $K_S$ over $K$. In this case theorem 4 is a reformulation of classical Galois theory.
In particular, if $K$ is a finite field, then $\pi_1(Spec K) \simeq \hat \mathbb{Z}$.
Let $X = Spec A$, where $A$ is the ring of integers in an algebraic number field $K$. Let $N$ be the maximal algebraic extension of $N$ that is unramified at all non-zero prime ideals of $A$. Then $\pi_1(X)$ is the Galois group of $N$ over $K$.
More generally, if $a \in A$, $a \neq 0$, then $\pi_1(Spec A[\frac{1}{a}])$ is the Galois group, over $K$, of the maximal algebraic extension of $K$ that is unramified at all non-zero prime ideals of $A$ not dividing $a$.
In this section we explain the connection between the main theorem of Galois theory for schemes, theorem 4, and classical Galois theory.
We denote by $k$ a field. It is our purpose to show that the opposite category of the category of free separable $K$-algebras is equivalent to the category of finite $\pi_1(X)$-sets, for a certain profinite group $\pi_1(X)$. This is a special case of the main theorem 4, with $X = Spec K$. In the general proof we shall use the contents of this section only for algebraically closed $K$. In that case, which is much simpler, the group $\pi_1(X)$ is trivial, so that the category of finite $\pi_1(X)$-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, $\mathcal{C}$, satisfying a small number of properties together with a fibre functor $F: \mathcal{C}\to FinSet$, preserving those properties. The theory is more fully described in the entry on Grothendieck's Galois theory.
One notices that classical Galois theory has an equivalent reformulation in topos theory. That puts it into a wider general abstract context and leads to a topos-theoretic general Galois theory.
The étale morphisms $f : Y \to X$ corresponds precisely to the locally constant sheaves on $X$ with respect to the etale topology, in that it is equivalently a morphism for which there is an etale cover $\{U_i \to X\}$ such that $f$ is a constant sheaf on each $U_i$.
For $K$ a field let $Et(K)$ be its small étale site. And
the sheaf topos over it. This topos is a
Then Galois extensions of $K$ correspond precisely to the locally constant objects in $\mathcal{E}$. The full subcategory on them is the Galois topos $Gal(\mathcal{E}) \hookrightarrow \mathcal{E}$.
The Galois group is the fundamental group of the topos.
Accordingly in topos theory Galois theory is generally about the classification of locally constant sheaves. The Galois group corresponds to 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, $Gal(\overline{\mathbb{Q}},\mathbb{Q})$ 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, $X$ is contained in the étale fundamental group of $X$? 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)
geometric homotopy groups in an (∞,1)-topos, fundamental group of a topos, fundamental ∞-groupoid of an (∞,1)-topos
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
Galois theory in a presentable symmetric monoidal stable (infinity,1)-category is developed in
See also
Tamás Szamuely, Galois groups and fundamental groups, Cambridge Studies in Adv. Math.