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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 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 :''K'' = Q(,) 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() and Q(), is Q(). L-function Biquadratic fields are the simplest examples of abelian extensions of Q that are not cyclic extensions. According to general theory the Dedekind zeta-function of such a field is a product of the Riemann zeta-function and three Dirichlet L-functions. Those L-functions are for the Dirichlet characters which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
