On Dirichlet biquadratic fields

Abstract

We prove the existence of a subset, with positive natural density, of squarefree integers n>0 such that the 4–rank of the ideal class group of ℚ(-n,n) is ω 3 (n)-1, where ω 3 (n) is the number of prime divisors of n that are 3 modulo 4. Recall that for the class groups associated to ℚ(n) or ℚ(-n) an analogous subset of n does not exist.