Galois extension
English
Etymology
Named for its connection with Galois theory and after French mathematician Évariste Galois.
Noun
Galois extension (plural Galois extensions)
- (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.
- The significance of a Galois extension is that it has a Galois group and obeys the fundamental theorem of Galois theory.
- 1986, D. J. H. Garling, A Course in Galois Theory, Cambridge University Press, page 108,
- Corollary If is a Galois extension, there exists an irreducible polynomial in such that is a splitting field extension for over .
- 1989, Katsuya Miyake, On central extensions, Jean-Marie De Koninck, Claude Levesque (editors), Number Theory, Walter de Gruyter, page 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.
- 2003, Paul M. Cohn, Basic Algebra: Groups, Rings and Fields, Springer, page 211,
- 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
Derived terms
Related terms
Translations
algebraic extension that is normal and separable
|
Further reading
Galois group on Wikipedia.Wikipedia
Normal extension on Wikipedia.Wikipedia
Separable extension on Wikipedia.Wikipedia
Degree of a field extension on Wikipedia.Wikipedia
Splitting field on Wikipedia.Wikipedia
- Galois extension on Encyclopedia of Mathematics