Meaning of Galois extension | Babel Free

Definitions

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.

Equivalents

Examples

“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.”

“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.”

“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.”

“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.”

CEFR level