Galois group

# Galois group

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Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, more specifically in the area of modern algebra known as Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...

, the Galois group of a certain type of field extension
Field extension
In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties...

is a specific group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

associated with the field extension. The study of field extensions (and polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

s which give rise to them) via Galois groups is called Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...

, so named in honor of Évariste Galois
Évariste Galois
Évariste Galois was a French mathematician born in Bourg-la-Reine. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long-standing problem...

who first discovered them.

For a more elementary discussion of Galois groups in terms of permutation groups, see the article on Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...

.

## Definition

Suppose that E is an extension
Field extension
In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties...

of the field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

F (written as E/F and read E over F). An automorphism
Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the 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
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations.  If there exists an isomorphism between two structures, the two structures are said to be isomorphic.  In a certain sense, isomorphic structures are...

α from E to E such that α(x) = x for each x in F. The set of all automorphisms of E/F forms a group with the operation of function composition
Function composition
In mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...

. This group is sometimes denoted by Aut(E/F).

If E/F is a Galois extension
Galois extension
In mathematics, a Galois extension is an algebraic field extension E/F satisfying certain conditions ; one also says that the extension is Galois. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.The definition...

, then Aut(E/F) is called the Galois group of (the extension) E over F, and is usually denoted by Gal(E/F).

## Examples

In the following examples F is a field, and C, R, Q are the fields of complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

, real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

, and rational
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

numbers, respectively. The notation F(a) indicates the field extension
Field extension
In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties...

In abstract algebra, adjunction is a construction in field theory, where for a given field extension E/F, subextensions between E and F are constructed.- Definition :...

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.
• Aut(R/Q) is trivial. Indeed it can be shown that any Q-automorphism must preserve the ordering
Order theory
Order theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and gives some basic definitions...

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(³√2). The group Aut(K/Q) contains only the identity automorphism. This is because K is not a normal extension
Normal extension
In abstract algebra, an algebraic field extension L/K is said to be normal if L is the splitting field of a family of polynomials in K[X]...

, since the other two cube roots of 2 (both complex) are missing from the extension — in other words K is not a splitting field
Splitting field
In abstract algebra, a splitting field of a polynomial with coefficients in a field is a smallest field extension of that field over which the polynomial factors into linear factors.-Definition:...

.
• Consider now L = Q(³√2, ω), where ω is a primitive third root of unity
Root of unity
In mathematics, a root of unity, or de Moivre number, is any complex number that equals 1 when raised to some integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, field theory, and the discrete...

. The group Gal(L/Q) is isomorphic to S3, the dihedral group of order 6
Dihedral group of order 6
The smallest non-abelian group has 6 elements. It is a dihedral group with notation D3 and the symmetric group of degree 3, with notation S3....

, 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
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...

of order q and qn respectively, then Gal(E/F) is cyclic of order n.
• If f is an irreducible polynomial
Irreducible polynomial
In mathematics, the adjective irreducible means that an object cannot be expressed as the product of two or more non-trivial factors in a given set. See also factorization....

of prime degree p with rational coefficients and exactly two non-real roots, then the Galois group of f is the full symmetric group
Symmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...

Sp.

## Properties

The significance of an extension being Galois is that it obeys the fundamental theorem of Galois theory
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 which is finite and Galois, there is a one-to-one correspondence between its...

: the closed (with respect to the Krull topology below) 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
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

, called the Krull topology, that makes it into a profinite group.