Galois extension

Template:Short description In mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable;Template:Sfn 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.Template:Efn

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.Template:Sfn

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

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 $E_i\subset E_j$.Template:Sfn

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.

References

• Template:Lang Algebra

Further reading

• (Galois' original paper, with extensive background and commentary.)
• Template:Springer
• (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".)