Galois group

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In mathematics, a Galois group is a group associated with a certain type of field extension. The study of field extensions (and polynomials which give rise to them) via Galois groups is called Galois theory.

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

Definition of the Galois group

Suppose that E is an extension of the field F. Consider the set of all field automorphisms of E/F; that is, isomorphisms α from E to itself, such that α(x) = x for every x in F. This set of automorphisms with the operation of function composition forms a group G, sometimes denoted Aut(E/F).

If E/F is a Galois extension, then G is called the Galois group of the extension, and is usually denoted Gal(E/F).

Examples

In the following examples F is a field, and <math>\mathbb{C}</math>, <math>\mathbb{R}</math>, and <math>\mathbb{Q}</math> are the fields of complex, real, and rational numbers, respectively. A field <math>F(a)</math> is the field extension obtained by adjoining an element a to the field F.

• <math>Gal(F/F)</math> is the trivial group that has a single element.
• <math>Gal(\mathbb{C}/\mathbb{R})</math> has two elements, the identity automorphism and the complex conjugation automorphism.
• <math>Gal(\mathbb{Q}(\sqrt{2})/\mathbb{Q})</math> has two elements, the identity automorphism and the automorphism which exchanges √2 and −√2.
• <math>Aut(\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q})</math> contains only the trivial identity. This is because <math>\mathbb{Q}(\sqrt[3]{2})</math> is not a normal extension, since the other two cube roots of 2 (both complex) are missing from the extension (In other words E is not a splitting field).
• <math>Gal\left(\mathbb{Q}(\sqrt[3]{2}, \omega)/\mathbb{Q}\right)</math>, where ω is a primitive third root of unity, is isomorphic to S₃, the dihedral group of order 6.
• <math>Aut(\mathbb{R}/\mathbb{Q})</math> contains only the identity (it can be shown that a Q-automorphism must preserve the order of the reals and hence must be the identity).
• <math>Gal\left(\mathbb{C}/\mathbb{Q}\right)</math> is an infinite group.

Facts

The significance of an extension being Galois is that it obeys the fundamental theorem of Galois theory: the subgroups of the Galois group correspond to the intermediate fields of the field extension.

It can be shown that E is algebraic over F if and only if the Galois group is pro-finite.de:Galoisgruppe es:Grupo de Galois fr:Groupe de Galois