Galois group

From Example Problems
Jump to navigation Jump to search

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.

de:Galoisgruppe es:Grupo de Galois fr:Groupe de Galois