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Bi‐Quadratic Number Fields with Trivial 2‐Primary Hilbert Kernels
Author(s) -
Kolster Manfred,
Movahhedi Abbas
Publication year - 2003
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611502014028
Subject(s) - mathematics , quadratic equation , algebraic number field , kernel (algebra) , pure mathematics , genus , conjecture , quadratic field , discrete mathematics , quadratic function , botany , biology , geometry
Using results of Browkin and Schinzel one can easily determine quadratic number fields with trivial 2‐primary Hilbert kernels. In the present paper we completely determine all bi‐quadratic number fields which have trivial 2‐primary Hilbert kernels. To obtain our results, we use several different tools, amongst which is the genus formula for the Hilbert kernel of an arbitrary relative quadratic extension, which is of independent interest. For some cases of real bi‐quadratic fields there is an ambiguity in the genus formula, so in this situation we use instead Brauer relations between the Dedekind zeta‐funtions and the Birch–Tate conjecture. 2000 Mathematics Subject Classification 11R70, 19F15.