In mathematics, a biquadratic field is a number field of a particular kind, which is a Galois extension of the rational number field with Galois group isomorphic to the Klein four-group.
Biquadratic fields are all obtained by adjoining two square roots. Therefore in explicit terms they have the form
for rational numbers and . There is no loss of generality in taking and to be non-zero and square-free integers.
According to Galois theory, there must be three quadratic fields contained in , since the Galois group has three subgroups of index 2. The third subfield, to add to the evident and , is .
Biquadratic fields are the simplest examples of abelian extensions of that are not cyclic extensions.