biquadratic extension

A biquadratic extension of a field $F$ is a Galois extension $K$ of $F$ such that $\operatorname{Gal} (K/F)$ 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 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 $F$ .