Imaginary Quadratic

is called the 2-class field tower of k. If n is the minimal integer such that kn = kn+1, then n is called the length of the tower. If no such n exists, then the tower is said to be of infinite length. At present there is no known decision procedure to determine whether or not the (2-)class field tower of a given field k is infinite. However, it is known by group theoretic results (see [2]) that if rankCl2(k) ≤ 2, then the tower is finite, in fact of length at most 3. (Here the rank means minimal number of generators.) On the other hand, until now (see Table 1 and the penultimate paragraph of this introduction) all examples in the mathematical literature of imaginary quadratic fields with rankCl2(k) ≥ 3 (let us mention in particular Schmithals [13]) have infinite 2-class field tower. Nevertheless, if we are interested in developing a decision procedure for determining if the 2-class field tower of a field is infinite, then a good starting point would be to find a procedure for sieving out those fields with rankCl2(k) ≤ 2. We have already started this program for imaginary quadratic number fields k. In [1] we classified all imaginary quadratic fields whose 2-class field k1 has cyclic 2-class group. In this paper we determine when Cl2(k) has rank 2 for imaginary quadratic fields k with Cl2(k) of type (2, 2m). (The notation (2, 2m) means the direct sum of a group of order 2 and a cyclic group of order 2m.) The group theoretic results mentioned above also show that such fields have 2-class field tower of length 2.