# Galois group

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*). The significance of an extension being Galois is that it obeys the fundamental theorem of Galois theory.

It can be shown that *E* is algebraic
over *F* if and only if the Galois group is pro-finite.

## Examples

- If
*E*=*F*, then the Galois group is the trivial group that has a single element. - If
*F*is the field**R**of real numbers, and*E*is the field**C**of complex numbers, then the Galois group has two elements, namely the identity automorphism and the complex conjugation automorphism. - If
*F*is**Q**(the field of rational numbers), and*E*is**Q**(√2), the field obtained from**Q**by adjoining √2, then the Galois group again has two elements: the identity automorphism, and the automorphism which exchanges √2 and −√2. - If
*F*is**Q**, and*E*is**Q**(α), where α is the real cube root of 2, then*E*/*F*is not a Galois extension. This is because it is not a normal extension, since the other two cube roots of 2, being complex numbers, are not contained in**Q**(α). In other words*E*is not a splitting field. There is no automorphism of*E*apart from the identity. - If
*F*is**Q**and*E*is the field of real numbers, then the automorphism group is trivial: the only automorphism of*E*is the identity. - If
*F*is**Q**and*E*is the field of complex numbers, then the Galois group is infinite.