Author(s)

Daniel C. Mayer

Naglergasse 53, 8010 Graz, Austria.

Let p be a prime and K be a number field with non-trivial p-class group Cl_pK. 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=G²_pk, that is the second derived quotient M≃G/Gⁿ. The family τ₁K of abelian type invariants of the p-class groups Cl_pL 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 (τ₁Kτ⁽²⁾K) includes data on unramified abelian extensions L/K of degree p² and enables sharper bounds for the order of M in the case Cl_pk≃(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 p², or even p³, and admits the identification of the p-class tower group G for various infinite series of quadratic fields K=Q(√d) with Cl_pK≃(p,p) possessing a p-class field tower of exact length l_pK=3 as a striking novelty.

Hilbert p-Class Field Tower, p-Class Group, p-Principalization Types, Quadratic Fields, Unramified Cyclic Cubic Field Extensions, p-Class Tower Group, Relation Rank, Metabelianization, Coclass Graphs

Mayer, D. (2017) Criteria for Three-Stage Towers of p-Class Fields. Advances in Pure Mathematics, 7, 135-179. doi: 10.4236/apm.2017.72008.