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# The $2$-rank of the class group of imaginary bicyclic biquadratic fields

Published:1997-04-01
Printed: Apr 1997
• Thomas M. McCall
• Charles J. Parry
• Ramona R. Ranalli
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## Abstract

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 $Z_2$-matrix determined by Legendre symbols involving pairs of ramified primes. As applications, all subfields with both $2$-class and class group $Z_2 \times Z_2$ are determined. The final results assume the completeness of D.~A.~Buell's list of imaginary fields with small class numbers.
 MSC Classifications: 11R16 - Cubic and quartic extensions 11R29 - Class numbers, class groups, discriminants 11R20 - Other abelian and metabelian extensions

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