Criteria for Three-Stage Towers of <i>p</i>-Class Fields

Abstract

Let p be a prime and K be a number field with non-trivial p -class group Cl p K . A crucial step in identifying the Galois group G ∞ p of the maximal unramified pro- p extension of K is to determine its two-stage approximation M=G 2 p k, that is the second derived quotient M &simeq; G/G n . The family τ 1 K of abelian type invariants of the p -class groups Cl p L of all unramified cyclic extensions L/K of degree p is called the index- abelianization data (IPAD) of K . It is able to specify a finite batch of contestants for the second p -class group M of K . In this paper we introduce two different kinds of generalized IPADs for obtaining more sophisticated results. The multi-layered IPAD ( τ 1 K τ (2) K ) includes data on unramified abelian extensions L/K of degree p 2 and enables sharper bounds for the order of M in the case Cl p k &simeq; (p,p,p) , where current im-plementations of the p -group generation algorithm fail to produce explicit contestants for M , due to memory limitations. The iterated IPAD of second order τ (2) K contains information on non-abelian unramified extensions L/K of degree p 2 , or even p 3 , and admits the identification of the p -class tower group G for various infinite series of quadratic fields K= Q (√ d ) with Cl p K &simeq; (p,p) possessing a p -class field tower of exact length l p K=3 as a striking novelty.