Galois extension

In mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable; or equivalently, E/F is algebraic, and the field fixed by the automorphism group Aut(E/F) is precisely the base field F. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.

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 with fixed field F, then E/F is a Galois extension.

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 extension and a separable extension.
• E is a splitting field of a separable polynomial with coefficients in F.
• |Aut(E/F)| = [E:F], that is, the number of automorphisms equals the degree of the extension.

Other equivalent statements are:

• Every irreducible polynomial in F[x] with at least one root in E splits over E and is separable.
• |Aut(E/F)| ≥ [E:F], that is, the number of automorphisms is at least the degree of the extension.
• F is the fixed field of a subgroup of Aut(E).
• F is the fixed field of Aut(E/F).
• There is a one-to-one correspondence between subfields of E/F and subgroups of Aut(E/F).

Examples

There are two basic ways to construct examples of Galois extensions.

• Take any field E, any subgroup of Aut(E), and let F be the fixed field.
• Take any field F, any separable polynomial in F[x], and let E be its splitting field.

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 closure 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. For more detailed examples, see the page on the fundamental theorem of Galois theory

An algebraic closure $\bar K$ of an arbitrary field $K$ is Galois over $K$ if and only if $K$ is a perfect field.

See also

• (Galois' original paper, with extensive background and commentary.)
• (Chapter 4 gives an introduction to the field-theoretic approach to Galois theory.)
• (This book introduces the reader to the Galois theory of Grothendieck, and some generalisations, leading to Galois groupoids.)
• . English translation (of 2nd revised edition): (Later republished in English by Springer under the title "Algebra".)