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

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 Z2-matrix determined by Legendre symbols involving pairs of ramified primes. As applications, all subfields with both 2-class and class group Z2 × Z2 are determined. The final results assume the completeness of D. A. Buell's list of imaginary fields with small class numbers.