Galois Field 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 $F.$
*, \!\operatorname(E/F), = :F that is, the number of automorphisms equals the degree of the extension.

Other equivalent statements are:

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

Examples

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

* Take any field $E$, any finite subgroup of $\operatorname(E)$, and let $F$ be the fixed field.
* Take any field $F$, any separable polynomial in F /math>, 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 $\bar K$ of an arbitrary field $K$ is Galois over $K$ if and only if $K$ is a perfect field.

Notes

Citations

References

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Further reading

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* ''(Galois' original paper, with extensive background and commentary.)''
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* ''(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.)
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*. English translation (of 2nd revised edition): ''(Later republished in English by Springer under the title "Algebra".)''
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{{DEFAULTSORT:Galois Extension
Galois theory
Algebraic number theory
Field extensions