Galois extension

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

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

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

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

• differential Galois extension

• Hopf-Galois extension

• 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