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.
- H |-> EH