Determination of all imaginary quadratic fields for which their Hilbert 2-class fields have 2-class groups of rank 2

We determine those imaginary quadratic number fields whose 2-class groups have rank 3 and 4-rank <= 2 and such that the 2-class groups of their Hilbert 2-class fields have rank 2. This result, along with previous work, gives a complete determination of all complex quadratic number fields for which the 2-class groups of their 2-class fields have rank at most 2. (C) 2021 Elsevier Inc. All rights reserved.