TITLE:
Successive Approximation of p-Class Towers
AUTHORS:
Daniel C. Mayer
KEYWORDS:
p-Class Towers, Galois Groups, Second p-Class Groups, Abelian Type Invariants of p-Class Groups, p-Transfer Kernel Types, Artin Limit Pattern, Quadratic Fields, Unramified Cyclic Extensions of Degree p, Dihedral Fields of Degree 2p, Finite p-Groups, Maximal Nilpotency Class, Maximal Subgroups, Polycyclic Pc-Presentations, Commutator Calculus, Central Series
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.7 No.12,
December
21,
2017
ABSTRACT: 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 C1pE 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 : C1pF → C1pE 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.