Galois extension

From Wiktionary, the free dictionary
Jump to navigation Jump to search

English

[edit]
English Wikipedia has an article on:
Wikipedia

Etymology

[edit]

Named for its connection with Galois theory and after French mathematician Évariste Galois.

Noun

[edit]

Galois extension (plural Galois extensions)

  1. (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.
    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

[edit]
  • 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

[edit]

Derived terms

[edit]
[edit]

Translations

[edit]

Further reading

[edit]