  Let F be a number field and p be a prime. In the Successive Approximation
Theorem, we prove that, for each positive integer n, finitely many candidates
for the Galois group G(p,n,F) of the n-th stage F(p,n) of the p-class tower
F(p,infinity) over F are determined by abelian type invariants of p-class
groups Cl(p,E) of unramified extensions E/F with degree [E:F]=$p^{n-1}$.
Illustrated by the most extensive numerical results available currently, the
transfer kernels ker(T(F,E)) of the p-class extensions T(F,E):Cl(p,F)-->Cl(p,E)
from F to unramified cyclic degree-p extensions E/F are shown to be capable of
narrowing down the number of contestants significantly. By determining the
isomorphism type of the maximal subgroups S<G of all 3-groups G with coclass
cc(G)=1, and establishing a general theorem on the connection between the
p-class towers of a number field F and of an unramified abelian p-extensions
E/F, we are able to provide a theoretical proof of the realization of certain
3-groups S with maximal class by 3-tower groups G(3,infinity,E) of dihedral
fields E with degree 6, which could not be realized up to now.
