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- Biquadratic_field abstract "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. Such fields are all obtained by adjoining two square roots. 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 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. 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 are the Jacobi symbols 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 the conductor of a field. Such L-functions have applications in analytic theory (Siegel zeroes), and in some of Kronecker's work.".
- Biquadratic_field wikiPageID "3718379".
- Biquadratic_field wikiPageLength "2136".
- Biquadratic_field wikiPageOutDegree "24".
- Biquadratic_field wikiPageRevisionID "656906322".
- Biquadratic_field wikiPageWikiLink Abelian_extension.
- Biquadratic_field wikiPageWikiLink Algebraic_number_field.
- Biquadratic_field wikiPageWikiLink Category:Algebraic_number_theory.
- Biquadratic_field wikiPageWikiLink Category:Galois_theory.
- Biquadratic_field wikiPageWikiLink Conductor_(class_field_theory).
- Biquadratic_field wikiPageWikiLink Dedekind_zeta_function.
- Biquadratic_field wikiPageWikiLink Dirichlet_L-function.
- Biquadratic_field wikiPageWikiLink Dirichlet_character.
- Biquadratic_field wikiPageWikiLink Galois_extension.
- Biquadratic_field wikiPageWikiLink Galois_group.
- Biquadratic_field wikiPageWikiLink Galois_theory.
- Biquadratic_field wikiPageWikiLink Jacobi_symbol.
- Biquadratic_field wikiPageWikiLink Klein_four-group.
- Biquadratic_field wikiPageWikiLink Leopold_Kronecker.
- Biquadratic_field wikiPageWikiLink Mathematics.
- Biquadratic_field wikiPageWikiLink Quadratic_field.
- Biquadratic_field wikiPageWikiLink Rational_number.
- Biquadratic_field wikiPageWikiLink Riemann_zeta_function.
- Biquadratic_field wikiPageWikiLink Siegel_zero.
- Biquadratic_field wikiPageWikiLink Square-free_integer.
- Biquadratic_field wikiPageWikiLink Square_root.
- Biquadratic_field wikiPageWikiLink Subgroup.
- Biquadratic_field wikiPageWikiLink Without_loss_of_generality.
- Biquadratic_field wikiPageWikiLinkText "Biquadratic field".
- Biquadratic_field wikiPageWikiLinkText "biquadratic field".
- Biquadratic_field wikiPageUsesTemplate Template:Citation.
- Biquadratic_field wikiPageUsesTemplate Template:Citation_needed.
- Biquadratic_field subject Category:Algebraic_number_theory.
- Biquadratic_field subject Category:Galois_theory.
- Biquadratic_field hypernym K.
- Biquadratic_field type School.
- Biquadratic_field comment "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. Such fields are all obtained by adjoining two square roots. 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 and square-free integers.".
- Biquadratic_field label "Biquadratic field".
- Biquadratic_field sameAs Q4915520.
- Biquadratic_field sameAs m.09w_v5.
- Biquadratic_field sameAs Q4915520.
- Biquadratic_field wasDerivedFrom Biquadratic_field?oldid=656906322.
- Biquadratic_field isPrimaryTopicOf Biquadratic_field.