Biquadratic field

Template:Short description In mathematics, a biquadratic field is a number field Template:Mvar of a particular kind, which is a Galois extension of the rational number field Template:Mvar with Galois group isomorphic to the Klein four-group.

Structure and subfields

Biquadratic fields are all obtained by adjoining two square roots. Therefore in explicit terms they have the form

$K=\mathbb{Q}(\sqrt{a},\sqrt{b})$

for rational numbers Template:Mvar and Template:Mvar. There is no loss of generality in taking Template:Mvar and Template:Mvar to be non-zero and square-free integers.

According to Galois theory, there must be three quadratic fields contained in Template:Mvar, since the Galois group has three subgroups of index 2. The third subfield, to add to the evident Template:Math and Template:Math, is Template:Math.

Biquadratic fields are the simplest examples of abelian extensions of Template:Math that are not cyclic extensions.

References

• Section 12 of Template:Citation