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. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In abstract algebra, an extension of a field K is a field L which contains K as a subfield. ... In mathematics, a Galois group is a group associated with a certain type of field extension. ... In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions. ...

The definition is as follows. The extension E/F is Galois if 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.) In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, a Galois group is a group associated with a certain type of field extension. ...

A fundamental result of Galois theory states that a finite extension E/F is Galois if and only if either of the following conditions holds: In mathematics, Galois theory is a branch of abstract algebra. ... In abstract algebra, an extension of a field K is a field L which contains K as a subfield. ...

• E/F is both a normal extension and a separable extension.
• More concretely, that E is the splitting field of a separable polynomial with coefficients in F.

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. 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]. The following conditions are equivalent to L/K being a normal extension: Let Ka an algebraic closure of K containing L. Every... In mathematics, a separable extension of a field K is a field L containing K that can be generated by adjoining to K a set of elements α, each of which is a root of a separable polynomial over K. In that case, each β in L has a separable... In abstract algebra, the splitting field of a polynomial P(X) over a given field K is a field extension L of K, over which P factorizes into linear factors X - ai, and such that the ai generate L over K. It can be shown that such splitting fields exist... In mathematics, an irreducible polynomial P(X) is separable if its roots in an algebraic closure are distinct - that is P(X) has distinct linear factors in some large enough field extension. ... Emil Artin (March 3, 1898-December 20, 1962) was an Austrian mathematician born in Vienna who spent his career in Germany (mainly in Hamburg) until the Nazi threat when he emigrated to the USA in 1937 where he was at Indiana University 1938-1946, and Princeton University 1946-1958. ...