3-class field towers with 2 or 3 stages

For quadratic fields k=Q(d^1/2) with discriminant d, 3-class group Cl(3,k) =(Z/3Z)^2, and one of six principalization types kappa(k) in {(1122),(2122),(3122),(1231),(2231),(4231)}, we seek necessary and sufficient conditions for the Galois group S=Gal(F(3,infinity,k)/k) of the unramified Hilbert 3-class field tower of k to coincide with the Galois group M=Gal(F(3,2,k)/k) of the maximal metabelian unramified 3-extension of k.In the case of non-coincidence,we study the path between M and S in the descendant tree of the elementary bicyclic 3-group (Z/3Z)^2. Minimal discriminants d with assigned principalization type kappa(k) and fixed length ell(3,k) in {2,3} of the 3-class field tower are determined experimentally for nilpotency class 5 <= cl(M) <= 8. 2000 Mathematics Subject Classification. 11R37, 11R29, 11R11, 11R16, 11R20; 20D15, 20F14.