nLab Galois theory

see also at Galois group

Idea

Classical Galois theory classifies field extensions. It is a special case of a classification of locally constant sheaves in a topos by permutation 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.

Classical Galois theory for fields

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.

Galois theory of fields

Definition

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 $G$.

Definition

If $K \subset L$ is a Galois extension, we define the Galois group to be

$$Gal(L/K) := Aut_K(L) \,.$$

This means that we have

$$K \simeq L^{Gal(L/K)} \,.$$

Definition

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 subextension $L \subset \bar K$ is called separable over $K$ if each $\alpha \in L$ is separable over $K$.

Definition

We call $L$ normal over $K$ if for each $\alpha \in L$ the polynomial $f^\alpha_K$ splits completely in linear factors in $L[X]$.

Theorem

Let $K$ be a field and $L$ a subfield of $\bar K$

$$K \subset L \subset \bar K \,.$$

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:

1. $L$ is a Galois extension of $K$.

2. $L$ is normal and separable over $K$.

3. There is a set $F \subset K[X] - \{0\}$ of separable polynomials such that $L$ is the splitting field of $F$ over $K$.

4. $\coprod_{E \in I} E \simeq L$

If these conditions are satisfied, then there is a group isomorphism

$$Gal(L/K) \simeq {\lim_{\leftarrow}}_{E \in I} Gal(E/K) \,,$$

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.

Theorem

(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

$$\{E | E\;is\;a\;subfield\;of\;L\;containing\;K\} \stackrel{\overset{\phi}{\to}}{\underset{\psi}{\leftarrow}} \{H|H\;is\;a\;closed\;subgroup\;of\;G\}$$

defined by

$$\phi(E) = Aut_E(L)$$

and

$$\psi(H) = L^H$$

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

1. $K \subset E$ is finite precisely if $H$ is open (in the profinite topology on $G$)

   $[E:K] \simeq index[G:H]$ if $H$ is open;

2. $E \subset L$ is Galois with $Gal(L/E) \simeq H$ (as topological groups);

3. for every $\sigma \in G$ we have that $\sigma[E]$ corresponds to $\sigma H \sigma^{-1}$;

4. $K \subset E$ 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.

In terms of separable algebras

Free modules

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

$$f(w_i) = \sum_{j = 1}^n a_{i j} w_j \;\;\; (1 \leq i \leq n)$$

for certain $a_{i j} \in A$, and the trace $Tr(f)$ of $f$ is defined by

$$Tr(f) = \sum_{i = 1}^n a_{i i} \,.$$

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.

Separable algebras

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

$$\phi(x) : y \mapsto Tr(x y) \,,$$

for $x, y \in B$.

Definition

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.

Separable closure

Recall the notion of separable elements

Definition

Let $K$ be a field and $\bar K$ an algebraic closure of $K$. The separable closure $K_S$ of $K$ is defined by

$$K_S \simeq \{x \in \bar K | x \; is \; separable \; over \; K\} \,.$$

Remark

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.

Definition

The Galois group $Gal(K_S/K)$ is called the absolute Galois group of $K$.

Galois theory for separable algebras

Theorem

Let $K$ be a field and $\pi_1(Spec K)$ its absolute Galois group. Then there is an equivalence of categories

$$SAlg_K^{op} \simeq \pi_1(Spec K) Set \,.$$

Galois theory for schemes

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.

Statement of the main theorem

Definition

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 $X$.

Theorem

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.

Examples

• 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 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 $a$.

Reproducing classical Galois theory of field extensions

In this section we explain the connection between the main theorem of Galois theory for schemes, theorem , 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 , 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.

Theorem

(…)

Grothendieck’s Galois Theory

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.

Galois theory in topos 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.

Reformulation of classical Galois theory

Observation

The étale morphisms $f : Y \to X$ corresponds precisely to the locally constant sheaves on $X$ with respect to the étale topology, in that it is equivalently a morphism for which there is an étale 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

$$\mathcal{E} := Sh(Et(K))$$

the sheaf topos over it. This topos is a

• local topos;

• locally connected topos;

• connected topos.

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.

Topos-theoretic Galois theory

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 .

(…)

Higher topos theoretic Galois theory

In the context of higher topos theory, there are accordingly higher analogs of Galois theory. According to shape theory, any (∞,1)-topos $\mathbf{H}$ has an associated fundamental ∞-groupoid $\Pi(\mathbf{H})$, which in general is a pro-space whose $1$-truncation $\Pi_1(\mathbf{H})$ is the ordinary fundamental groupoid of the underlying 1-topos. Classical topos-theoretic Galois theory states that locally constant sheaves (of sets) on a locally connected topos are equivalent to representations of $\Pi_1(\mathbf{H})$, i.e., functors $\Pi_1(\mathbf{H})\to Set$. This generalizes to higher topoi as follows:

Theorem

Let $\mathbf{H}$ be a locally n-connected (n+1,1)-topos, $-1\leq n\leq \infty$. Then there is an equivalence of categories

$$\mathbf{H}^lc \simeq \Fun(\Pi_{n+1}(\mathbf{H}), n Grpd),$$

where $\mathbf{H}^{lc}\subset\mathbf{H}$ is the subcategory of locally constant objects.

This generalization of Galois theory is discussed in (Grothendieck 75, Hoyois 13, Hoyois 15). For further discussion in the case $n=\infty$, see

• cohesive (infinity,1)-topos – structures– Galois theory

‘La Longue Marche à travers la Théorie de Galois’

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 will (one day be) discussed at 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.

(…)

Related entries

• Galois group, motivic Galois group

• Galois connection

• categorical Galois theory, Grothendieck's Galois theory

• Galois cover

• differential Galois theory

• Galois cohomology

• Lascar group (a Galois group of first order theories)

• fundamental theorem of covering spaces

• anabelian geometry

• geometric homotopy groups in an (∞,1)-topos, fundamental group of a topos, fundamental ∞-groupoid of an (∞,1)-topos

References

Original articles:

• Michael Barr, Abstract Galois Theory, J. Pure Appl. Algebra 19 (1980) 21–42 [pdf, pdf]

• Michael Barr, Abstract Galois Theory II, J. Pure Appl. Algebra 25 3 (1982) 227–247 [doi:10.1016/0022-4049(82)90080-9, pdf, pdf]

Introduction:

• Hendrik Lenstra, Galois theory for schemes , Mathematisch Instituut Universiteit van Amsterdam (1985) [pdf]

  Some of the material above is taken from this.

• Ravi Vakil, Kirsten Wickelgren, Universal covering spaces and fundamental groups in algebraic geometry as schemes, (pdf).

Monographs:

• Francis Borceux, George Janelidze, Galois Theories, Camb. Stud. Adv. Math. 72, Cambridge: Cambridge University Press (2001)

• Jean-Pierre Tignol, Galois’ Theory of Algebraic Equations, World Scientific (2001) [doi:10.1142/4628]

The locally simply connected case is discussed for instance in

• Marco Robalo, Galois theory towards dessins d’enfants, dissertation, pdf

Galois theory in a presentable symmetric monoidal stable (infinity,1)-category is developed in

• Akhil Mathew, The Galois group of a stable homotopy theory, arXiv.

Galois theory in (higher) topos theory

• SGA4, Expose VIII, Proposition 2.1.

• Marc Hoyois, Higher Galois theory, J. Pure Appl. Alg. 222:7, (2018) 1859-1877 arXiv:1506.07155 doi

• Joseph Rennie, The Unreasonable Efficacy of the Lifting Condition in Higher Categorical Galois Theory I: a Quasi-categorical Galois Theorem [arXiv:2409.03347]

See also

• The Galois Theory Web Page

• Wikipedia: Galois theory

• Tamás Szamuely, Galois groups and fundamental groups, Cambridge Studies in Adv. Math. 117 (2009)

• Alexander Grothendieck, letter to Larry Breen 17/2/1975 pdf, scan

• Alexander Grothendieck, La Longue Marche a Travers la Théorie de Galois pdf

category: Galois theory