In mathematics, 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.

Definition Edit

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

If   is a Galois extension, then   is called the Galois group of  , and is usually denoted by  .[1]

If   is not a Galois extension, then the Galois group of   is sometimes defined as  , where   is the Galois closure of  .

Galois group of a polynomial Edit

Another definition of the Galois group comes from the Galois group of a polynomial  . If there is a field   such that   factors as a product of linear polynomials


over the field  , then the Galois group of the polynomial   is defined as the Galois group of   where   is minimal among all such fields.

Structure of Galois groups Edit

Fundamental theorem of Galois theory Edit

One of the important structure theorems from Galois theory comes from the fundamental theorem of Galois theory. This states that given a finite Galois extension  , there is a bijection between the set of subfields   and the subgroups   Then,   is given by the set of invariants of   under the action of  , so


Moreover, if   is a normal subgroup then  . And conversely, if   is a normal field extension, then the associated subgroup in   is a normal group.

Lattice structure Edit

Suppose   are Galois extensions of   with Galois groups   The field   with Galois group   has an injection   which is an isomorphism whenever  .[2]

Inducting Edit

As a corollary, this can be inducted finitely many times. Given Galois extensions   where   then there is an isomorphism of the corresponding Galois groups:


Examples Edit

In the following examples   is a field, and   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.

Computational tools Edit

Cardinality of the Galois group and the degree of the field extension Edit

One of the basic propositions required for completely determining the Galois groups[3] of a finite field extension is the following: Given a polynomial  , let   be its splitting field extension. Then the order of the Galois group is equal to the degree of the field extension; that is,


Eisenstein's criterion Edit

A useful tool for determining the Galois group of a polynomial comes from Eisenstein's criterion. If a polynomial   factors into irreducible polynomials   the Galois group of   can be determined using the Galois groups of each   since the Galois group of   contains each of the Galois groups of the  

Trivial group Edit

  is the trivial group that has a single element, namely the identity automorphism.

Another example of a Galois group which is trivial is   Indeed, it can be shown that any automorphism of   must preserve the ordering of the real numbers and hence must be the identity.

Consider the field   The group   contains only the identity automorphism. This is because   is not a normal extension, since the other two cube roots of  ,


are missing from the extension—in other words K is not a splitting field.

Finite abelian groups Edit

The Galois group   has two elements, the identity automorphism and the complex conjugation automorphism.[4]

Quadratic extensions Edit

The degree two field extension   has the Galois group   with two elements, the identity automorphism and the automorphism   which exchanges   and  . This example generalizes for a prime number  

Product of quadratic extensions Edit

Using the lattice structure of Galois groups, for non-equal prime numbers   the Galois group of   is


Cyclotomic extensions Edit

Another useful class of examples comes from the splitting fields of cyclotomic polynomials. These are polynomials   defined as


whose degree is  , Euler's totient function at  . Then, the splitting field over   is   and has automorphisms   sending   for   relatively prime to  . Since the degree of the field is equal to the degree of the polynomial, these automorphisms generate the Galois group.[5] If   then


If   is a prime  , then a corollary of this is


In fact, any finite abelian group can be found as the Galois group of some subfield of a cyclotomic field extension by the Kronecker–Weber theorem.

Finite fields Edit

Another useful class of examples of Galois groups with finite abelian groups comes from finite fields. If q is a prime power, and if   and   denote the Galois fields of order   and   respectively, then   is cyclic of order n and generated by the Frobenius homomorphism.

Degree 4 examples Edit

The field extension   is an example of a degree   field extension.[6] This has two automorphisms   where   and   Since these two generators define a group of order  , the Klein four-group, they determine the entire Galois group.[3]

Another example is given from the splitting field   of the polynomial


Note because   the roots of   are   There are automorphisms


generating a group of order  . Since   generates this group, the Galois group is isomorphic to  .

Finite non-abelian groups Edit

Consider now   where   is a primitive cube root of unity. The group   is isomorphic to S3, the dihedral group of order 6, and L is in fact the splitting field of   over  

Quaternion group Edit

The Quaternion group can be found as the Galois group of a field extension of  . For example, the field extension


has the prescribed Galois group.[7]

Symmetric group of prime order Edit

If   is an irreducible polynomial of prime degree   with rational coefficients and exactly two non-real roots, then the Galois group of   is the full symmetric group  [2]

For example,   is irreducible from Eisenstein's criterion. Plotting the graph of   with graphing software or paper shows it has three real roots, hence two complex roots, showing its Galois group is  .

Comparing Galois groups of field extensions of global fields Edit

Given a global field extension   (such as  ) and equivalence classes of valuations   on   (such as the  -adic valuation) and   on   such that their completions give a Galois field extension


of local fields, there is an induced action of the Galois group   on the set of equivalence classes of valuations such that the completions of the fields are compatible. This means if   then there is an induced isomorphism of local fields


Since we have taken the hypothesis that   lies over   (i.e. there is a Galois field extension  ), the field morphism   is in fact an isomorphism of  -algebras. If we take the isotropy subgroup of   for the valuation class  


then there is a surjection of the global Galois group to the local Galois group such that there is an isomorphism between the local Galois group and the isotropy subgroup. Diagrammatically, this means


where the vertical arrows are isomorphisms.[8] This gives a technique for constructing Galois groups of local fields using global Galois groups.

Infinite groups Edit

A basic example of a field extension with an infinite group of automorphisms is  , since it contains every algebraic field extension  . For example, the field extensions   for a square-free element   each have a unique degree   automorphism, inducing an automorphism in  

One of the most studied classes of infinite Galois group is the absolute Galois group, which is an infinite, profinite group defined as the inverse limit of all finite Galois extensions   for a fixed field. The inverse limit is denoted


where   is the separable closure of the field  . Note this group is a topological group.[9] Some basic examples include   and


Another readily computable example comes from the field extension   containing the square root of every positive prime. It has Galois group


which can be deduced from the profinite limit


and using the computation of the Galois groups.

Properties Edit

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   is a Galois extension, then   can be given a topology, called the Krull topology, that makes it into a profinite group.

See also Edit

Notes Edit

  1. ^ Some authors refer to   as the Galois group for arbitrary extensions   and use the corresponding notation, e.g. Jacobson 2009.
  2. ^ a b Lang, Serge. Algebra (Revised Third ed.). pp. 263, 273.
  3. ^ a b "Abstract Algebra" (PDF). pp. 372–377. Archived (PDF) from the original on 2011-12-18.
  4. ^ Cooke, Roger L. (2008), Classical Algebra: Its Nature, Origins, and Uses, John Wiley & Sons, p. 138, ISBN 9780470277973.
  5. ^ Dummit; Foote. Abstract Algebra. pp. 596, 14.5 Cyclotomic Extensions.
  6. ^ Since   as a   vector space.
  7. ^ Milne. Field Theory. p. 46.
  8. ^ "Comparing the global and local galois groups of an extension of number fields". Mathematics Stack Exchange. Retrieved 2020-11-11.
  9. ^ "9.22 Infinite Galois theory". The Stacks project.
  10. ^ Milne. "Field Theory" (PDF). p. 98. Archived (PDF) from the original on 2008-08-27.
  11. ^ "Infinite Galois Theory" (PDF). p. 14. Archived (PDF) from the original on 6 April 2020.

References Edit

External links Edit