Successive Approximation of p-Class Towers

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 n th stage of the p -class tower over F are determined by abelian type invariants of p -class groups C1 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 ( T E, F ) of the p -class extensions T E, F : C1 p F → C1 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 -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.