Let
F be a number field and p be a prime. In the successive approximation theorem, we prove that, for each integer
n ≥ 1, finitely many candidates for the Galois group
of the
nth stage
of the
p-class tower
over
F are determined by abelian type invariants of
p-class groups C1
pE of unramified extensions
E/F with degree [
E :
F] =
pn-1. Illustrated by the most extensive numerical results available currently, the transfer kernels (
TE, F) of the
p-class extensions
TE, F : C1
pF → C1
pE 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-extension
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
of dihedral fields
E with degree 6, which could not be realized up to now.