biquadratic extension

A biquadratic extension of a field F is a Galois extensionMathworldPlanetmath K of F such that Gal(K/F) is isomorphicPlanetmathPlanetmathPlanetmath to the Klein 4-group. It receives its name from the fact that any such K is the compositum of two distinct quadratic extensions of F. The name can be somewhat misleading, however, since biquadratic extensions of F have exactly three distinct subfieldsMathworldPlanetmath that are quadratic extensions of F. This is easily seen to be true by the fact that the Klein 4-group has exactly three distinct subgroupsMathworldPlanetmathPlanetmath of order ( 2.

Note that, if α,βF, then F(α,β) is a biquadratic extension of F if and only if none of α, β, and αβ are squares in F.

Title biquadratic extension
Canonical name BiquadraticExtension
Date of creation 2013-03-22 15:56:21
Last modified on 2013-03-22 15:56:21
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 7
Author Wkbj79 (1863)
Entry type Definition
Classification msc 11R16
Related topic BiquadraticField
Related topic BiquadraticEquation2