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Criteria for Three-Stage Towers of p-Class Fields

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DOI: 10.4236/apm.2017.72008    924 Downloads   1,277 Views Citations
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ABSTRACT

Let p be a prime and K be a number field with non-trivial p-class group ClpK. A crucial step in identifying the Galois group Gp of the maximal unramified pro-p extension of K is to determine its two-stage approximation M=G2pk, that is the second derived quotient MG/Gn. The family τ1K of abelian type invariants of the p-class groups ClpL 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 Clpk(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 ClpK(p,p) possessing a p-class field tower of exact length lpK=3 as a striking novelty.

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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.

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