A formula is obtained for the rank of the $2$-Sylow subgroup of theideal class group of imaginary bicyclic biquadratic fields. Thisformula involves the number of primes that ramify in the field, the ranks of the $2$-Sylow subgroups of the ideal class groups of thequadratic subfields and the rank of a $Z_2$-matrix determined byLegendre symbols involving pairs of ramified primes. Asapplications, all subfields with both $2$-class and class group$Z_2 imes Z_2$ are determined. The final results assume thecompleteness of D.~A.~Buell's list of imaginary fields with smallclass numbers.
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