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:
“The motive of science was the extension of man, on all sides, into Nature, till his hands should touch the stars, his eyes see through the earth, his ears understand the language of beast and bird, and the sense of the wind; and, through his sympathy, heaven and earth should talk with him. But that is not our science.”
—Ralph Waldo Emerson (1803–1882)