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In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory, so named in honor of Évariste Galois who first discovered them.

For a more elementary discussion of Galois groups in terms of permutation groups, see the article on Galois theory.


Definition

Suppose that E is an extension of the field F (written as E/F and read E over F). An automorphism of E/F is defined to be an automorphism of E that fixes F pointwise. In other words, an automorphism of E/F is an isomorphism α from E to E such that α(x) = x for each x in F. The set of all automorphisms of E/F forms a group with the operation of function composition. This group is sometimes denoted by Aut(E/F).

If E/F is a Galois extension, then Aut(E/F) is called the Galois group of (the extension) E over F, and is usually denoted by Gal(E/F).[1]

If E/F is not a Galois extension, then the Galois group of (the extension) E over F is sometimes defined as Aut(G/F), where G is the Galois closure of E.


Examples

In the following examples F is a field, and C, R, Q are the fields of complex, real, and rational numbers, respectively. The notation F(a) indicates the field extension obtained by adjoining an element a to the field F.

For a finite field \( \mathbf{F}_{q^n} \), we always have \( \operatorname{Gal}(\mathbf{F}_{q^n}/\mathbf{F}_{q}) \) cyclic of order n, generated by the qth power Frobenius automorphism.

Properties

The significance of an extension being Galois is that it obeys the fundamental theorem of Galois theory: the closed (with respect to the Krull topology) subgroups of the Galois group correspond to the intermediate fields of the field extension.

If E/F is a Galois extension, then Gal(E/F) can be given a topology, called the Krull topology, that makes it into a profinite group.
See also

Absolute Galois group

Notes

Some authors refer to Aut(E/F) as the Galois group for arbitrary extensions E/F and use the corresponding notation, e.g. Jacobson 2009.

Cooke, Roger L. (2008), Classical Algebra: Its Nature, Origins, and Uses, John Wiley & Sons, p. 138, ISBN 9780470277973.

References

Jacobson, Nathan (2009) [1985], Basic algebra I (Second ed.), Dover Publications, ISBN 978-0-486-47189-1
Lang, Serge (2002), Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556

External links

Hazewinkel, Michiel, ed. (2001), "Galois group", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
"Galois Groups" at MathPages.com.

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