Galois group

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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.

[edit] Definition

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).

[edit] Examples

In the following examples F is a field, and C, R, and 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) contains only the identity. Indeed it can be shown that any Q-automorphism must preserve the ordering of the real numbers and hence must be the identity.
• Gal(C/Q) is an infinite group.
• The group

has two elements, the identity automorphism and the automorphism which exchanges √2 and −√2.

• Consider the field

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

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.

[edit] Facts

The significance of an extension being Galois is that it obeys the fundamental theorem of Galois theory: the subgroups of the Galois group correspond to the intermediate fields of the field extension.

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