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.

The property of an extension being Galois behaves well with respect to field composition and intersection.

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 $F.$
• $|\mathrm{Aut}(E/F)|=[E:F],$ 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.
• $|\mathrm{Aut}(E/F)|\ge [E:F],$ that is, the number of automorphisms is at least the degree of the extension.
• $F$ is the fixed field of a subgroup of $\mathrm{Aut}(E).$
• $F$ is the fixed field of $\mathrm{Aut}(E/F).$
• There is a one-to-one correspondence between subfields of $E/F$ and subgroups of $\mathrm{Aut}(E/F).$

An infinite field extension $E/F$ is Galois if and only if $E$ is the union of finite Galois subextensions $E_i/F$ indexed by an (infinite) index set $I$, i.e. $E=\bigcup _{i\in I}{E}_i$ and the Galois group is an inverse limit $\mathrm{Aut}(E/F)=\underset{i\in I}{\underleftarrow{\lim }}\mathrm{Aut}(E_i/F)$ where the inverse system is ordered by field inclusion $E_i\subset E_j$.

Examples

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

• Take any field $E$, any finite subgroup of $\mathrm{Aut}(E)$, 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 cubic 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 $x^2-2$; the second has normal closure that includes the complex cubic 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 $x^3-2$ has just one real root. For more detailed examples, see the page on the fundamental theorem of Galois theory.

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

Further reading

• (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".)