Galois extension
Encyclopedia
In mathematics
, a Galois extension is an algebraic field extension
E/F satisfying certain conditions (described below); 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 is as follows. An algebraic field extension E/F is Galois if it is normal
and separable
. Equivalently, the extension E/F is Galois if and only if
it is algebraic
, and the field fixed by the automorphism
group Aut(E/F) is precisely the base field F. (See the article Galois group
for definitions of some of these terms and some examples.)
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, then E/F is a Galois extension, where F is the fixed field of G.
that includes the complex cube roots of unity, and so is not a splitting field. In fact, it has no automorphism other than the identity, because it is contained in the real numbers and X3 − 2 has just one real root.
An algebraic closure
of an arbitrary field 
is Galois over 
if and only if 
is a perfect field
.
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...
, a Galois extension is an algebraic field extension
Algebraic extension
In abstract algebra, a field extension L/K is called algebraic if every element of L is algebraic over K, i.e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i.e...
E/F satisfying certain conditions (described below); one also says that the extension is Galois. The significance of being a Galois extension is that the extension has a Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
and 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 definition is as follows. An algebraic field extension E/F is Galois if it is normal
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]...
and separable
Separable extension
In modern algebra, an algebraic field extension E\supseteq F is a separable extension if and only if for every \alpha\in E, the minimal polynomial of \alpha over F is a separable polynomial . Otherwise, the extension is called inseparable...
. Equivalently, the extension E/F is Galois if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
it is algebraic
Algebraic extension
In abstract algebra, a field extension L/K is called algebraic if every element of L is algebraic over K, i.e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i.e...
, and the field fixed by the 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...
group Aut(E/F) is precisely the base field F. (See the article Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
for definitions of some of these terms and some examples.)
A result of Emil Artin
Emil Artin
Emil Artin was an Austrian-American mathematician of Armenian descent.-Parents:Emil Artin was born in Vienna to parents Emma Maria, née Laura , a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of Armenian descent...
allows one to construct Galois extensions as follows: If E is a given field, and G is a finite group of automorphisms of E, then E/F is a Galois extension, where F is the fixed field of G.
Characterization of Galois extensions
An important theorem of Emil Artin states that for a finite extension E/F, each of the following statements is equivalent to the statement that E/F is Galois:- E/F is a normal extensionNormal extensionIn 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]...
and a separable extensionSeparable extensionIn modern algebra, an algebraic field extension E\supseteq F is a separable extension if and only if for every \alpha\in E, the minimal polynomial of \alpha over F is a separable polynomial . Otherwise, the extension is called inseparable...
. - E is a splitting fieldSplitting fieldIn 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:...
of a separable polynomialSeparable polynomialIn mathematics, two slightly different notions of separable polynomial are used, by different authors.According to the most common one, a polynomial P over a given field K is separable if all its roots are distinct in an algebraic closure of K, that is the number of its distinct roots is equal to...
with coefficients in F. - [E:F] = |Aut(E/F)|; that is, the degree of the field extension is equal to the orderOrder (group theory)In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....
of the automorphism group of E/F.
Examples
Adjoining to the rational number field the square root of 2 gives a Galois extension, while adjoining the cube root of 2 gives a non-Galois extension. Both these extensions are separable, because they have characteristic zero. The first of them is the splitting field of X2 − 2; the second has normal closureNormal 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]...
that includes the complex cube roots of unity, and so is not a splitting field. In fact, it has no automorphism other than the identity, because it is contained in the real numbers and X3 − 2 has just one real root.
An algebraic closure
Algebraic closure
In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics....
of an arbitrary field is Galois over if and only if is a perfect fieldPerfect field
In algebra, a field k is said to be perfect if any one of the following equivalent conditions holds:* Every irreducible polynomial over k has distinct roots.* Every polynomial over k is separable.* Every finite extension of k is separable...
.