Galois group

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.

are missing from the extension—in other words K is not a splitting field.

In fact, any finite abelian group can be found as the Galois group of some subfield of a cyclotomic field extension by the Kronecker–Weber theorem.

then there's a surjection of the global Galois group to the local Galois group such that there's and isomorphism between the local Galois group and the isotropy subgroup. Diagrammatically, this means

where the vertical arrows are isomorphisms.[8] This gives a technique for constructing Galois groups of local fields using global Galois groups.

The significance of an extension being Galois is that it obeys the fundamental theorem of Galois theory: the closed (with respect to the Krull topology) subgroups of the Galois group correspond to the intermediate fields of the field extension.