Galois extension

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

Noun

 * 1)  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.

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) $$F$$.

Translations

 * French: extension de Galois, extension galoisienne
 * Italian: estensione di Galois