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The 2-Rank of the Class Group of Imaginary Bicyclic Biquadratic Fields

Published online by Cambridge University Press:  20 November 2018

Thomas M. McCall ,
Charles J. Parry  and
Ramona R. Ranalli
Thomas M. McCall
Affiliation:
Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, 24061-0123
Charles J. Parry
Affiliation:
Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, 24061-0123
Ramona R. Ranalli
Affiliation:
Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, 24061-0123
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Abstract

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A formula is obtained for the rank of the 2-Sylow subgroup of the ideal class group of imaginary bicyclic biquadratic fields. This formula involves the number of primes that ramify in the field, the ranks of the 2-Sylow subgroups of the ideal class groups of the quadratic subfields and the rank of a Z2 -matrix determined by Legendre symbols involving pairs of ramified primes. As applications, all subfields with both 2- class and class group Z 2×Z 2 are determined. The final results assume the completeness of D. A. Buell’s list of imaginary fields with small class numbers.

Keywords

11R1611R2911R20

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 49 , Issue 2 , 01 April 1997 , pp. 283 - 300
Copyright
Copyright © Canadian Mathematical Society 1997

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