- Biquadratic field
In
mathematics , a biquadratic field is anumber field "K" of a particular kind, which is aGalois extension of therational number field Q withGalois group theKlein four-group . Such fields are all obtained by adjoining twosquare root s. Therefore in explicit terms we have:"K" = Q(√"a",√"b")
for rational numbers "a" and "b". There is no
loss of generality in taking "a" and "b" to be non-zero andsquare-free integer s.According to
Galois theory , there must be threequadratic field s contained in "K", since the Galois group has threesubgroup s 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 extension s of Q that are notcyclic extension s. According to general theory theDedekind zeta-function of such a field is a product of theRiemann zeta-function and threeDirichlet L-function s. Those L-functions are for theDirichlet character s which are theJacobi symbol s attached to the three quadratic fields. Therefore taking the product of the Dedekind zeta-functions of the quadratic fields, multiplying them together, and dividing by the square of the Riemann zeta-function, is a recipe for the Dedekind zeta-function of the biquadratic field. This illustrates also some general principles on abelian extensions, such as the calculation of theconductor of a field .Such L-functions have applications in analytic theory (
Siegel zero es), and in some ofKronecker 's work.
Wikimedia Foundation. 2010.
