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Concept
Galois group
Summary
In mathematics, in the area of abstract 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 their relationship to the polynomials that 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.
Suppose that is an extension of the field (written as and read "E over F "). An automorphism of is defined to be an automorphism of that fixes pointwise. In other words, an automorphism of is an isomorphism such that for each . The set of all automorphisms of forms a group with the operation of function composition. This group is sometimes denoted by
If is a Galois extension, then is called the Galois group of , and is usually denoted by .
If is not a Galois extension, then the Galois group of is sometimes defined as , where is the Galois closure of .
Another definition of the Galois group comes from the Galois group of a polynomial . If there is a field such that factors as a product of linear polynomials
over the field , then the Galois group of the polynomial is defined as the Galois group of where is minimal among all such fields.
One of the important structure theorems from Galois theory comes from the fundamental theorem of Galois theory. This states that given a finite Galois extension , there is a bijection between the set of subfields and the subgroups Then, is given by the set of invariants of under the action of , so
Moreover, if is a normal subgroup then . And conversely, if is a normal field extension, then the associated subgroup in is a normal group.
Suppose are Galois extensions of with Galois groups The field with Galois group has an injection which is an isomorphism whenever .
As a corollary, this can be inducted finitely many times.
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