Galois extension

plural Galois extensions

1. Galois theoryalgebraAn algebraic extension that is both a normal and a separable extension; equivalently, an algebraic extension E/F such that the fixed field of its automorphism group (Galois group) Aut(E/F) is the base field F.

   The significance of a Galois extension is that it has a Galois group and obeys the fundamental theorem of Galois theory.

   4 more examples

   The fundamental theorem of Galois theory states that there is a one-to-one correspondence between the subfields of a Galois extension and the subgroups of its Galois group.

   Corollary If L#58;K is a Galois extension, there exists an irreducible polynomial f in K#91;x#93; such that L#58;K is a splitting field extension for f over K.

   1986, D. J. H. Garling, A Course in Galois Theory, Cambridge University Press, page 108

   First, arithmetic obstructions against constructing central extensions of a fixed finite base Galois extension are analyzed with the local-global principle to give some quantitative estimates of them.

   1989, Katsuya Miyake, “On central extensions”, in Jean-Marie De Koninck, Claude Levesque, editors, Number Theory, Walter de Gruyter, page 642

   With the help of the results in Section 7.5 it is not hard to describe all Galois extensions. Proposition 7.6.1. Let E#47;F be a finite field extension. Then (i) E#47;F is a Galois extension if and only if it is normal and separable; (ii) E#47;F is contained in a Galois extension if and only if it is separable.

   2003, Paul M. Cohn, Basic Algebra: Groups, Rings and Fields, Springer, page 211