Biquadratic field

Template:Short description In mathematics, a biquadratic field is a number field $K$ of a particular kind, which is a Galois extension of degree 4 of the rational number field $\mathbb{Q}$ with Galois group isomorphic to the Klein four-group.

Structure and subfields

Biquadratic fields are obtained by adjoining two square roots of rational numbers $a,b$:

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

Without loss of generality, we may assume a and b are non-zero square-free integers.

The Galois group has three subgroups of index 2, corresponding to the three quadratic subfields

$\mathbb{Q}(\sqrt{a}),\mathbb{Q}(\sqrt{b}),\mathbb{Q}(\sqrt{ab})\subset K$.

Biquadratic fields are the simplest examples of abelian extensions of $\mathbb{Q}$ that are not cyclic extensions. Also biquadratic fields are usually not monogenic: although there exists a primitive element which generates the field $K$ over $\mathbb{Q}$, its ring of integers ${𝒪}_K$ might not possess a single generator over $\mathbb{Z}$.

References

• Section 12 of Template:Citation