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 extensionsEdit

An important theorem of Emil Artin states that for a finite extension <math>E/F,</math> each of the following statements is equivalent to the statement that <math>E/F</math> is Galois:

<math>E/F</math> is a normal extension and a separable extension.
<math>E</math> is a splitting field of a separable polynomial with coefficients in <math>F.</math>
<math>|\!\operatorname{Aut}(E/F)| = [E:F],</math> that is, the number of automorphisms equals the degree of the extension.

Other equivalent statements are:

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

An infinite field extension <math>E/F</math> is Galois if and only if <math>E</math> is the union of finite Galois subextensions <math>E_i/F</math> indexed by an (infinite) index set <math>I</math>, i.e. <math>E=\bigcup_{i\in I}E_i</math> and the Galois group is an inverse limit <math>\operatorname{Aut}(E/F)=\varprojlim_{i\in I}{\operatorname{Aut}(E_i/F)}</math> where the inverse system is ordered by field inclusion <math>E_i\subset E_j</math>.Template:Sfn

ExamplesEdit

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

Take any field <math>E</math>, any finite subgroup of <math>\operatorname{Aut}(E)</math>, and let <math>F</math> be the fixed field.
Take any field <math>F</math>, any separable polynomial in <math>F[x]</math>, and let <math>E</math> 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 <math>x^2 -2</math>; 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 <math>x^3 -2</math> has just one real root. For more detailed examples, see the page on the fundamental theorem of Galois theory.

An algebraic closure <math>\bar K</math> of an arbitrary field <math>K</math> is Galois over <math>K</math> if and only if <math>K</math> is a perfect field.

NotesEdit

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CitationsEdit

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ReferencesEdit

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