Criteria for three-stage towers of p-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=G(p,K) of the maximal unramified pro-p extension of K is to determine its two-stage approximation M=G(p,2,K), that is the second derived quotient M=G/G". The family tau(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-p 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 ((tau(1,K),tau(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)=(p,p,p), where current implementations of the p-group generation algorithm fail to produce explicit contestants for M, due to memory limitations. The iterated IPAD of second order tau^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(squareroot(d)) with Cl(p,K)=(p,p) possessing a p-class field tower of exact length L(p,K)=3 as a striking novelty.