Galois extension en · NOUN
Etymology
Named for its connection with Galois theory and after French mathematician Évariste Galois.
Meanings
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(qualifier: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.
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.
1989, Katsuya Miyake, “On central extensions”, in Jean-Marie De Koninck, Claude Levesque, editors, Number Theory, Walter de Gruyter, page 642:The significance of a Galois extension is that it has a Galois group and obeys the fundamental theorem of Galois theory.
Corollary If L#58;K is a Galois extension, there exists an irreducible polynomial f in K#91;x#93; such that L#58;K is a splitting field extension for f over K.
1986, D. J. H. Garling, A Course in Galois Theory, Cambridge University Press, page 108:With the help of the results in Section 7.5 it is not hard to describe all Galois extensions. Proposition 7.6.1. Let E#47;F be a finite field extension. Then (i) E#47;F is a Galois extension if and only if it is normal and separable; (ii) E#47;F is contained in a Galois extension if and only if it is separable.
2003, Paul M. Cohn, Basic Algebra: Groups, Rings and Fields, Springer, page 211:
Forms
| Spelling | Features | Labels | Source |
|---|---|---|---|
| Galois extensions | Number=Plur | lexicographic |
Deriveds
Hypernyms
algebraic extension · normal extension · separable extension
Relateds
Galois group · Hopf-Galois extension
Translations (4)
pl rozszerzenie Galois (algebraic extension that is normal and separable) · fr extension de Galois (algebraic extension that is normal and separable) · it estensione di Galois (algebraic extension that is normal and separable) · fr extension galoisienne (algebraic extension that is normal and separable)