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.

Suppose E is an extension of the field F, and consider the set of all field automorphisms of E which fix F pointwise. This set of automorphisms forms a group G. If there are no elements of E \\ F which are fixed by all members of G, then the extension E/F is called a Galois extension, and G is the Galois group of the extension and is usually denoted Gal(E/F).

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 of real numbers, and E is the field of complex numbers, then the Galois group has 2 elements.
• If F is Q (the field of rational numbers), and E is Q(√2), then the Galois group again has 2 elements.
• If F is Q, and E is Q(the real cube root of 2), then the Galois group has 1 element. (This is because the other two cube roots of 2 are complex.)
• If F is Q and E is the real numbers, then the Galois group has 1 element.

Fundamental theorem of Galois theory. Let E be a finite Galois extension of the field F with Galois group G. For every subgroup H of G, let E^H denote the subfield of E consisting of all elements which are fixed by all elements of H. Then the function

H |-> E^H

is a bijection between the set of subgroups of G and the set of subfields of E that contain F. This function is monotone decreasing and its inverse is given by the Galois group of E/E^H. Furthermore, the field E^H is a normal extension of F if and only if H is a normal subgroup of G. If H is a normal subgroup of G then the restriction of G's elements to E^H induces an isomorphism between the group and the Galois group of the extension E^H/F.