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
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
.
If is not a Galois extension, then the Galois group of
is sometimes defined as
, where
is the of
.
Galois group of a polynomial
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
Fundamental theorem of Galois theory
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
Suppose are Galois extensions of
with Galois groups
The field
with Galois group
has an injection
which is an isomorphism whenever
.
Inducting
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
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
Cardinality of the Galois group and the degree of the field extension
One of the basic propositions required for completely determining the Galois groups 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
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
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
,
and
are missing from the extension—in other words K is not a (splitting field).
Finite abelian groups
The Galois group has two elements, the identity automorphism and the complex conjugation automorphism.
Quadratic extensions
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
Using the lattice structure of Galois groups, for non-equal prime numbers the Galois group of
is
Cyclotomic extensions
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. 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
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
The field extension is an example of a degree
field extension. 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.
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
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
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.
Symmetric group of prime order
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
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
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. This gives a technique for constructing Galois groups of local fields using global Galois groups.
Infinite groups
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. 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
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
- Fundamental theorem of Galois theory
- (Absolute Galois group)
- Galois representation
- (Demushkin group)
- Solvable group
Notes
- Some authors refer to
as the Galois group for arbitrary extensions
and use the corresponding notation, e.g. Jacobson 2009.
- Lang, Serge. Algebra (Revised Third ed.). pp. 263, 273.
- "Abstract Algebra" (PDF). pp. 372–377. (PDF) from the original on 2011-12-18.
- Cooke, Roger L. (2008), Classical Algebra: Its Nature, Origins, and Uses, John Wiley & Sons, p. 138, ISBN .
- Dummit; Foote. Abstract Algebra. pp. 596, 14.5 Cyclotomic Extensions.
- Since
as a
vector space.
- Milne. Field Theory. p. 46.
- "Comparing the global and local galois groups of an extension of number fields". Mathematics Stack Exchange. Retrieved 2020-11-11.
- "9.22 Infinite Galois theory". The Stacks project.
- Milne. "Field Theory" (PDF). p. 98. (PDF) from the original on 2008-08-27.
- "Infinite Galois Theory" (PDF). p. 14. (PDF) from the original on 6 April 2020.
References
- Jacobson, Nathan (2009) [1985]. Basic Algebra I (2nd ed.). Dover Publications. ISBN .
- Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN , MR 1878556