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Galois extension. In DICTIOUS you will not only get to know all the dictionary meanings for the word
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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 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.
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”, in 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