Galois extension

English Etymology Named for its connection with Galois theory and after French mathematician Évariste Galois. Noun en algebra Galois theory An 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. en The significance of a Galois extension is that it has a Galois group and obeys the fundamental theorem of Galois theory. en 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. en 1986 D. J. H. Garling A Course in Galois Theory https://books.google.com.au/books?id=g65XLke1wMMC&pg=PA108&dq=%22Galois+extension%22%7C%22Galois+extensions%22&hl=en&sa=X&ved=0ahUKEwje3vmiworfAhVDIJAKHWhtDi8Q6AEIqwIwMQ#v=onepage&q=%22Galois%20extension%22%7C%22Galois%20extensions%22&f=false 108 w:Cambridge University Press Corollary If is a Galois extension, there exists an irreducible polynomial in such that is a splitting field extension for over . en 1989 Katsuya Miyake On central extensions Jean-Marie De Koninck; Claude Levesque Number Theory Walter de Gruyter https://books.google.com.au/books?id=N_jaIH7KV7kC&pg=PA642&dq=%22Galois+extension%22%7C%22Galois+extensions%22&hl=en&sa=X&ved=0ahUKEwje3vmiworfAhVDIJAKHWhtDi8Q6AEIhAIwKg#v=onepage&q=%22Galois%20extension%22%7C%22Galois%20extensions%22&f=false 642 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. en 2003 Paul Cohn Paul M. Cohn Basic Algebra: Groups, Rings and Fields https://books.google.com.au/books?id=VESm0MJOiDQC&pg=PA211&dq=%22Galois+extension%22%7C%22Galois+extensions%22&hl=en&sa=X&ved=0ahUKEwje3vmiworfAhVDIJAKHWhtDi8Q6AEIrAEwGQ#v=onepage&q=%22Galois%20extension%22%7C%22Galois%20extensions%22&f=false 211 Springer With the help of the results in Section 7.5 it is not hard to describe all Galois extensions.Proposition 7.6.1. Let be a finite field extension. Then (i) is a Galois extension if and only if it is normal and separable; (ii) is contained in a Galois extension if and only if it is separable. Usage notes Given an algebraic extension of finite degree, the following conditions are equivalent: is both a normal extension and a separable extension. is a splitting field of some separable polynomial with coefficients in . ; that is, the number of automorphisms equals the degree of the extension. Every irreducible polynomial in with at least one root in splits over and is a separable polynomial. The fixed field of is exactly (instead of merely containing) . Hypernyms en algebraic extension en normal extension en separable extension Derived terms en differential Galois extension Related terms en Galois group en Hopf-Galois extension Translations algebraic extension that is normal and separable French: fr extension de Galois f, fr extension galoisienne f Italian: it estensione di Galois f Polish: pl rozszerzenie Galois n Further reading Galois group Normal extension Separable extension Degree of a field extension Splitting field Galois extension on Encyclopedia of Mathematics en eponyms