Galois extension

English

Etymology

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

Noun

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 $L:K$ is a Galois extension, there exists an irreducible polynomial $f$ in $K[x]$ such that $L:K$ is a splitting field extension for $f$ over $K$.

   • 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 $E/F$ be a finite field extension. Then (i) $E/F$ is a Galois extension if and only if it is normal and separable; (ii) $E/F$ is contained in a Galois extension if and only if it is separable.

Usage notes

• Given an algebraic extension $E/F$ of finite degree, the following conditions are equivalent:
  • $E/F$ is both a normal extension and a separable extension.
  • $E$ is a splitting field of some separable polynomial with coefficients in $F$.
  • $\vert \operatorname {Aut} (E/F)\vert =[E:F]$; that is, the number of automorphisms equals the degree of the extension.
  • Every irreducible polynomial in $F[x]$ with at least one root in $E$ splits over $E$ and is a separable polynomial.
  • The fixed field of $\operatorname {Aut} (E/F)$ is exactly (instead of merely containing) $F$.

Hypernyms

• algebraic extension
• normal extension
• separable extension

Derived terms

• differential Galois extension

Related terms

• Galois group
• Hopf-Galois extension

Translations

algebraic extension that is normal and separable

• French: extension de Galois f, extension galoisienne f
• Italian: estensione di Galois f

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