Biquadratic field

In mathematics, a biquadratic field is a number field K of a particular kind, which is a Galois extension of the rational number field ℚ 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

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

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

According to Galois theory, there must be three quadratic fields contained in K, since the Galois group has three subgroups of index 2. The third subfield, to add to the evident ℚ(√a) and ℚ(√b), is ℚ(√ab).

Biquadratic fields are the simplest examples of abelian extensions of ℚ that are not cyclic extensions.

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