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.
Highlights
For a prime number p and an algebraic number field F, let Fp(∞) be the p-class tower, more precisely the unramified Hilbert p-class field tower, that is the maximal unramified pro-p extension, of F
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 Cl p E of unramified extensions E F with degree [E : F ] = pn−1
Fp(n) F purpose with o, f this paper is to report on the most up-to-date theoretical view of p-class towers and the state of the art of actual numerical investigations
Summary
For a prime number p and an algebraic number field F, let Fp(∞) be the p-class tower, more precisely the unramified Hilbert p-class field tower, that is the maximal unramified pro-p extension, of F. ( ) are described by the derived quotients G / G(n) ( ) n ≥ 1 , of the p-class tower group= G : G= ∞p F : Gal. Fp(n) F purpose with o, f this paper is to report on the most up-to-date theoretical view of p-class towers and the state of the art of actual numerical investigations.
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