Biquadratic field
{{Short description|Object from algebraic number theory}}
In mathematics, a biquadratic field is a number field {{mvar|K}} of a particular kind, which is a Galois extension of the rational number field {{mvar|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
:
for rational numbers {{mvar|a}} and {{mvar|b}}. There is no loss of generality in taking {{mvar|a}} and {{mvar|b}} to be non-zero and square-free integers.
According to Galois theory, there must be three quadratic fields contained in {{mvar|K}}, since the Galois group has three subgroups of index 2. The third subfield, to add to the evident {{math|1=ℚ({{radic|a}})}} and {{math|1=ℚ({{radic|b}})}}, is {{math|1=ℚ({{radic|ab}})}}.
Biquadratic fields are the simplest examples of abelian extensions of {{math|1=ℚ}} that are not cyclic extensions.
References
- Section 12 of {{Citation
| last=Swinnerton-Dyer
| first=H.P.F.
| author-link=Peter Swinnerton-Dyer
| title=A brief guide to algebraic number theory
| publisher=Cambridge University Press
| year=2001
| isbn=978-0-521-00423-7
| series=London Mathematical Society Student Texts
| volume=50
| mr=1826558
}}
{{DEFAULTSORT:Biquadratic Field}}