On the 3-rank of the ideal class group of quadratic fields

Abstract

As for Scholz' inequalities s ≤ r ≤ s + 1 with respect to the 3-rank of the ideal class group of quadratic fields, we give a criterion to be r = s + 1. From this, we give a new family of imaginary quadratic fields whose ideal class groups have 3-rank at least two.