biquadratic extension

A biquadratic extension of a field $F$ is a Galois extension $K$ of $F$ such that $\mathrm{Gal}(K/F)$ is isomorphic 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 subfields 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 subgroups of order (http://planetmath.org/OrderGroup) 2.

Note that, if $\alpha ,\beta \in F$, then $F(\sqrt{\alpha },\sqrt{\beta })$ is a biquadratic extension of $F$ if and only if none of $\alpha$, $\beta$, and $\alpha \beta$ 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