In mathematics, a Galois extension is an algebraic field extension E/F satisfying certain conditions (described below); one also says that the extension is Galois. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.
The definition is as follows. An algebraic field extension E/F is Galois if it is normal and separable. Equivalently, the extension E/F is Galois if and only if it is algebraic, and the field fixed by the automorphism group Aut(E/F) is precisely the base field F. (See the article Galois group for definitions of some of these terms and some examples.)
A result of Emil Artin allows one to construct Galois extensions as follows: If E is a given field, and G is a finite group of automorphisms of E, then E/F is a Galois extension, where F is the fixed field of G.
Read more about Galois Extension: Characterization of Galois Extensions, Examples
Famous quotes containing the word extension:
“Where there is reverence there is fear, but there is not reverence everywhere that there is fear, because fear presumably has a wider extension than reverence.”
—Socrates (469–399 B.C.)