Galois group

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In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions (and polynomials which give rise to them) via Galois groups is called Galois theory, so named in honor of Évariste Galois who first discovered them.

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

Contents

Definition

Suppose that E is an extension of the field F (written as E/F and read E over F). Consider the set of all 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, sometimes denoted by Aut(E/F).

If E/F is a Galois extension, then Aut(E/F) is called the Galois group of (the extension) E over F, and is usually denoted by Gal(E/F).[1]

Examples

In the following examples F is a field, and C, R, Q are the fields of complex, real, and rational numbers, respectively. The notation F(a) indicates the field extension obtained by adjoining an element a to the field F.

  • Gal(F/F) is the trivial group that has a single element, namely the identity automorphism.
  • Gal(C/R) has two elements, the identity automorphism and the complex conjugation automorphism.
  • Aut(R/Q) is trivial. Indeed it can be shown that any Q-automorphism must preserve the ordering of the real numbers and hence must be the identity.
  • Aut(C/Q) is an infinite group.
  • Gal(Q(√2)/Q) has two elements, the identity automorphism and the automorphism which exchanges √2 and −√2.
  • Consider the field K = Q(³√2). The group Aut(K/Q) contains only the identity automorphism. This is because K is not a normal extension, since the other two cube roots of 2 (both complex) are missing from the extension — in other words K is not a splitting field.
  • Consider now L = Q(³√2, ω), where ω is a primitive third root of unity. The group Gal(L/Q) is isomorphic to S3, the dihedral group of order 6, and L is in fact the splitting field of x3 − 2 over Q.
  • If q is a prime power, and if F = GF(q) and E = GF(qn) denote the Galois fields of order q and qn respectively, then Gal(E/F) is cyclic of order n.

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