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

http://dx.doi.org/10.4153/CJM-1997-014-2
Canad. J. Math. 49(1997), 283-300

  Published:1997-04-01
 Printed: Apr 1997

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.