Galois group

In mathematics, more specifically in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory, so named in honor of Évariste Galois who first discovered them.

For a more elementary discussion of Galois groups in terms of permutation groups, see the article on Galois theory.


Suppose that E is an extension of the field F (written as E/F and read "E over F"). An automorphism of E/F is defined to be an automorphism of E that fixes F pointwise. In other words, an automorphism of E/F is an isomorphism α from E to E such that α(x) = x for each xF. The set of all automorphisms of E/F forms a group with the operation of function composition. This group is sometimes denoted by Aut(E/F).

If E/F is a Galois extension, then Aut(E/F) is called the "Galois group of (the extension) E over F, and is usually denoted by Gal(E/F).[1]

If E/F is not a Galois extension, then the Galois group of (the extension) E over F is sometimes defined as Aut(G/F), where G is the Galois closure of E.


In the following examples F is a field, and C, R, Q are the fields of complex, real, and rational numbers, respectively. The notation F(a) indicates the field extension obtained by adjoining an element a to the field F.

  • Gal(F/F) is the trivial group that has a single element, namely the identity automorphism.
  • Gal(C/R) has two elements, the identity automorphism and the complex conjugation automorphism.[2]
  • Aut(R/Q) is trivial. Indeed, it can be shown that any automorphism of R must preserve the ordering of the real numbers and hence must be the identity.
  • Aut(C/Q) is an infinite group.
  • Gal(Q(2)/Q) has two elements, the identity automorphism and the automorphism which exchanges +2 and −2.
  • Consider the field K = Q(32). The group Aut(K/Q) contains only the identity automorphism. This is because K is not a normal extension, since the other two complex cube roots of 2 are missing from the extension—in other words K is not a splitting field.
  • Consider now L = Q(32, ω), where ω is a primitive cube root of unity. The group Gal(L/Q) is isomorphic to S3, the dihedral group of order 6, and L is in fact the splitting field of x3 − 2 over Q.
  • If q is a prime power, and if F = GF(q) and E = GF(qn) denote the Galois fields of order q and qn respectively, then Gal(E/F) is cyclic of order n and generated by the Frobenius homomorphism.
  • If f is an irreducible polynomial of prime degree p with rational coefficients and exactly two nonreal roots, then the Galois group of f is the full symmetric group Sp.


The significance of an extension being Galois is that it obeys the fundamental theorem of Galois theory: the closed (with respect to the Krull topology) subgroups of the Galois group correspond to the intermediate fields of the field extension.

If E/F is a Galois extension, then Gal(E/F) can be given a topology, called the Krull topology, that makes it into a profinite group.

See also


  1. ^ Some authors refer to Aut(E/F) as the Galois group for arbitrary extensions E/F and use the corresponding notation, e.g. Jacobson 2009.
  2. ^ Cooke, Roger L. (2008), Classical Algebra: Its Nature, Origins, and Uses, John Wiley & Sons, p. 138, ISBN 9780470277973.


External links

Abelian extension

In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvable if its Galois group is solvable, i.e., if the group can be decomposed into a series of normal extensions of an abelian group.

Every finite extension of a finite field is a cyclic extension.

Class field theory provides detailed information about the abelian extensions of number fields, function fields of algebraic curves over finite fields, and local fields.

There are two slightly different definitions of the term cyclotomic extension. It can mean either an extension formed by adjoining roots of unity to a field, or a subextension of such an extension. The cyclotomic fields are examples. A cyclotomic extension, under either definition, is always abelian.

If a field K contains a primitive n-th root of unity and the n-th root of an element of K is adjoined, the resulting Kummer extension is an abelian extension (if K has characteristic p we should say that p doesn't divide n, since otherwise this can fail even to be a separable extension). In general, however, the Galois groups of n-th roots of elements operate both on the n-th roots and on the roots of unity, giving a non-abelian Galois group as semi-direct product. The Kummer theory gives a complete description of the abelian extension case, and the Kronecker–Weber theorem tells us that if K is the field of rational numbers, an extension is abelian if and only if it is a subfield of a field obtained by adjoining a root of unity.

There is an important analogy with the fundamental group in topology, which classifies all covering spaces of a space: abelian covers are classified by its abelianisation which relates directly to the first homology group.

Absolute Galois group

In mathematics, the absolute Galois group GK of a field K is the Galois group of Ksep over K, where Ksep is a separable closure of K. Alternatively it is the group of all automorphisms of the algebraic closure of K that fix K. The absolute Galois group is well-defined up to inner automorphism. It is a profinite group.

(When K is a perfect field, Ksep is the same as an algebraic closure Kalg of K. This holds e.g. for K of characteristic zero, or K a finite field.)

Class field theory

In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of local fields (one-dimensional local fields) and "global fields" (one-dimensional global fields) such as number fields and function fields of curves over finite fields in terms of abelian topological groups associated to the fields. It also studies various arithmetic properties of such abelian extensions. Class field theory includes global class field theory and local class field theory.

The abelian topological group CK associated to such a field K is the multiplicative group of a local field or the idele class group of a global field.

One of the fundamental results of class field theory is a construction of a nontrivial reciprocity homomorphism, which acts from CK to the Galois group of the maximal abelian extension of the field K. The existence theorem of class field theory states that each open subgroup of finite index of CK is the image with respect to the norm map from the corresponding class field extension down to K.

The theory takes its name from the fact that it includes a one-to-one correspondence between finite abelian extensions of a fixed local or global field and appropriate open subgroup of finite index in CK. For example, in the case of number fields, the latter are classes of ideals of the field or open subgroups of the idele class group of the field; the Hilbert class field, which is the maximal unramified abelian extension of a number field, corresponds to a very special class of ideals.

A standard method since the 1930s is to develop local class field theory, which describes abelian extensions of completions of a local field, and then use it to construct global class field theory.

Origins of class field theory (reciprocity laws) can be traced to Gauss. Class field theory is the top achievement of algebraic number theory of the 20th century. There is a variety of presentations of class field theory, ranging from using Brauer groups or not using them, from using Galois cohomology or not using it, using features of characteristic zero or of positive characteristic.

There are two types of class field theories for number fields: (1) very explicit but restricted class field theory such as cyclotomic and CM which work over very special number fields using additional structures (roots of unity, torsion points of elliptic curves with CM), (2) general class field theory which works over any global field (any number field) and which follows different conceptual vision and which, remarkably, is simpler than the very explicit class field theory.

There are three main generalizations of class field theory: the Langlands program, higher class field theory, anabelian geometry, each leading to its own insights into key aspects of number theory.

Dessin d'enfant

In mathematics, a dessin d'enfant is a type of graph embedding used to study Riemann surfaces and to provide combinatorial invariants for the action of the absolute Galois group of the rational numbers. The name of these embeddings is French for a "child's drawing"; its plural is either dessins d'enfant, "child's drawings", or dessins d'enfants, "children's drawings".

A dessin d'enfant is a graph, with its vertices colored alternately black and white, embedded in an oriented surface that, in many cases, is simply a plane. For the coloring to exist, the graph must be bipartite. The faces of the embedding must be topological disks. The surface and the embedding may be described combinatorially using a rotation system, a cyclic order of the edges surrounding each vertex of the graph that describes the order in which the edges would be crossed by a path that travels clockwise on the surface in a small loop around the vertex.

Any dessin can provide the surface it is embedded in with a structure as a Riemann surface. It is natural to ask which Riemann surfaces arise in this way. The answer is provided by Belyi's theorem, which states that the Riemann surfaces that can be described by dessins are precisely those that can be defined as algebraic curves over the field of algebraic numbers. The absolute Galois group transforms these particular curves into each other, and thereby also transforms the underlying dessins.

For a more detailed treatment of this subject, see Schneps (1994) or Lando & Zvonkin (2004).

Differential Galois theory

In mathematics, differential Galois theory studies the Galois groups of differential equations.

Elkies trinomial curves

In number theory, the Elkies trinomial curves are certain hyperelliptic curves constructed by Noam Elkies which have the property that rational points on them correspond to trinomial polynomials giving an extension of Q with particular Galois groups.

One curve, C168, gives Galois group PSL(2,7) from a polynomial of degree seven, and the other, C1344, gives Galois group AL(8), the semidirect product of a 2-elementary group of order eight acted on by PSL(2, 7), giving a transitive permutation subgroup of the symmetric group on eight roots of order 1344.

The equation of the curve C168 is

The curve is a plane algebraic curve model for a Galois resolvent for the trinomial polynomial equation x7 + bx + c = 0. If there exists a point (x, y) on the (projectivized) curve, there is a corresponding pair (b, c) of rational numbers, such that the trinomial polynomial either factors or has Galois group PSL(2,7), the finite simple group of order 168. The curve has genus two, and so by Faltings theorem there are only a finite number of rational points on it. These rational points were proven by Nils Bruin using the computer program Kash to be the only ones on C168, and they give only four distinct trinomial polynomials with Galois group PSL(2,7): x7-7x+3 (the Trinks polynomial), (1/11)x7-14x+32 (the Erbach-Fisher-McKay polynomial) and two new polynomials with Galois group PSL(2,7),



On the other hand, the equation of curve C1344 is

Once again the genus is two, and by Faltings theorem the list of rational points is finite. It is thought the only rational points on it correspond to polynomials x8+16x+28, x8+576x+1008, 19453x8+19x+2 which have Galois group AL(8), and x8+324x+567, which comes from two different rational points and has Galois group PSL(2, 7) again, this time as the Galois group of a polynomial of degree eight.

Field arithmetic

In mathematics, field arithmetic is a subject that studies the interrelations between arithmetic properties of a field and its absolute Galois group.

It is an interdisciplinary subject as it uses tools from algebraic number theory, arithmetic geometry, algebraic geometry, model theory, the theory of finite groups and of profinite groups.

Fundamental theorem of Galois theory

In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions.

In its most basic form, the theorem asserts that given a field extension E/F that is finite and Galois, there is a one-to-one correspondence between its intermediate fields and subgroups of its Galois group. (Intermediate fields are fields K satisfying F ⊆ K ⊆ E; they are also called subextensions of E/F.)

Galois cohomology

In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group G associated to a field extension L/K acts in a natural way on some abelian groups, for example those constructed directly from L, but also through other Galois representations that may be derived by more abstract means. Galois cohomology accounts for the way in which taking Galois-invariant elements fails to be an exact functor.

Galois extension

In mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable; or equivalently, E/F is algebraic, and the field fixed by the automorphism group Aut(E/F) is precisely the base field F. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory. A result of Emil Artin allows one to construct Galois extensions as follows: If E is a given field, and G is a finite group of automorphisms of E with fixed field F, then E/F is a Galois extension.

Galois module

In mathematics, a Galois module is a G-module, with G being the Galois group of some extension of fields. The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring, but can also be used as a synonym for G-module. The study of Galois modules for extensions of local or global fields is an important tool in number theory.

Galois theory

In mathematics, Galois theory provides a connection between field theory and group theory. Using Galois theory, certain problems in field theory can be reduced to group theory, which is in some sense simpler and better understood.

The subject is named after Évariste Galois, who introduced it for studying the roots

of a polynomial and characterizing the polynomial equations that are solvable by radicals in terms of properties of the permutation group of their roots—an equation is solvable by radicals if its roots may be expressed by a formula involving only integers, nth roots and the four basic arithmetic operations.

The theory has been popularized (among mathematicians) and developed by Richard Dedekind, Leopold Kronecker and Emil Artin, and others, who, in particular, interpreted the permutation group of the roots as the automorphism group of a field extension.

Galois theory has been generalized to Galois connections and Grothendieck's Galois theory.

Inverse Galois problem

In Galois theory, the inverse Galois problem concerns whether or not every finite group appears as the Galois group of some Galois extension of the rational numbers Q. This problem, first posed in the early 19th century, is unsolved.

There are some permutation groups for which generic polynomials are known, which define all algebraic extensions of Q having a particular group as Galois group. These groups include all of degree no greater than 5. There also are groups known not to have generic polynomials, such as the cyclic group of order 8.

More generally, let G be a given finite group, and let K be a field. Then the question is this: is there a Galois extension field L/K such that the Galois group of the extension is isomorphic to G? One says that G is realizable over K if such a field L exists.

Kronecker–Weber theorem

In algebraic number theory, it can be shown that every cyclotomic field is an abelian extension of the rational number field Q, having Galois group of the form ( Z / n Z ) × {\displaystyle (\mathbb {Z} /n\mathbb {Z} )^{\times }} . The Kronecker–Weber theorem provides a partial converse: every finite abelian extension of Q is contained within some cyclotomic field. In other words, every algebraic integer whose Galois group is abelian can be expressed as a sum of roots of unity with rational coefficients. For example,


The theorem is named after Leopold Kronecker and Heinrich Martin Weber.

Langlands dual group

In representation theory, a branch of mathematics, the Langlands dual LG of a reductive algebraic group G (also called the L-group of G) is a group that controls the representation theory of G. If G is defined over a field k, then LG is an extension of the absolute Galois group of k by a complex Lie group. There is also a variation called the Weil form of the L-group, where the Galois group is replaced by a Weil group. The Langlands dual group is also often referred to as an L-group; here the letter L indicates also the connection with the theory of L-functions, particularly the automorphic L-functions. The Langlands dual was introduced by Langlands (1967) in a letter to A. Weil.

The L-group is used heavily in the Langlands conjectures of Robert Langlands. It is used to make precise statements from ideas that automorphic forms are in a sense functorial in the group G, when k is a global field. It is not exactly G with respect to which automorphic forms and representations are functorial, but LG. This makes sense of numerous phenomena, such as 'lifting' of forms from one group to another larger one, and the general fact that certain groups that become isomorphic after field extensions have related automorphic representations.

Motive (algebraic geometry)

In algebraic geometry, motives (or sometimes motifs, following French usage) is a theory proposed by Alexander Grothendieck in the 1960's to unify the vast array of similarly behaved cohomology theories such as singular cohomology, de Rham cohomology, etale cohomology, and crystalline cohomology. Philosophically, a 'motif' is the 'cohomology essence' of a variety.

In the formulation of Grothendieck for smooth projective varieties, a motive is a triple , where X is a smooth projective variety, is an idempotent correspondence, and m an integer, however, such a triple contains almost no information outside the context of Grothendieck's category of pure motives, where a morphism from to is given by a correspondence of degree . A more object focussed approach is taken by Pierre Deligne in Le Groupe Fondamental de la Droite Projective Moins Trois Points. In that article, a motive is a 'system of realisations'. That is, a tuple

consisting of modules

over the rings

respectively, various comparison isomorphisms

between the obvious base changes of these modules, filtrations , a -action on and a "Frobenius" automorphism of . This data is modeled on the cohomologies of a smooth projective -variety and the structures and compatibilities they admit, and gives an idea about what kind of information is contained a motive.

Picard–Vessiot theory

In differential algebra, Picard–Vessiot theory is the study of the differential field extension generated by the solutions of a linear differential equation, using the differential Galois group of the field extension. A major goal is to describe when the differential equation can be solved by quadratures in terms of properties of the differential Galois group. The theory was initiated by Émile Picard and Ernest Vessiot from about 1883 to 1904.

Kolchin (1973) and van der Put & Singer (2003) give detailed accounts of Picard–Vessiot theory.

Resolvent (Galois theory)

In Galois theory, a discipline within the field of abstract algebra, a resolvent for a permutation group G is a polynomial whose coefficients depend polynomially on the coefficients of a given polynomial p and has, roughly speaking, a rational root if and only if the Galois group of p is included in G. More exactly, if the Galois group is included in G, then the resolvent has a rational root, and the converse is true if the rational root is a simple root. Resolvents were introduced by Joseph Louis Lagrange and systematically used by Évariste Galois. Nowadays they are still a fundamental tool to compute Galois groups. The simplest examples of resolvents are

These three resolvents have the property of being always separable, which means that, if they have a multiple root, then the polynomial p is not irreducible. It is not known if there is an always separable resolvent for every group of permutations.

For every equation the roots may be expressed in terms of radicals and of a root of a resolvent for a resoluble group, because, the Galois group of the equation over the field generated by this root is resoluble.

Solvable group

In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup.

Historically, the word "solvable" arose from Galois theory and the proof of the general unsolvability of quintic equation. Specifically, a polynomial equation is solvable by radicals if and only if the corresponding Galois group is solvable.

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