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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} \left({\sqrt {a}},{\sqrt {b}}\right).

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 ${\mathcal {O}}_{K}$ might not possess a single generator over $\mathbb {Z}$ .

References

Section 12 of Swinnerton-Dyer, H.P.F. (2001), A brief guide to algebraic number theory, London Mathematical Society Student Texts, vol. 50, Cambridge University Press, ISBN 978-0-521-00423-7, MR 1826558