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Article citations


Blackburn, N. (1957) On Prime-Power Groups in Which the Derived Group Has Two Generators. Proceedings of the Cambridge Philosophical Society, 53, 19-27.

has been cited by the following article:

  • TITLE: Criteria for Three-Stage Towers of p-Class Fields

    AUTHORS: Daniel C. Mayer

    KEYWORDS: 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

    JOURNAL NAME: Advances in Pure Mathematics, Vol.7 No.2, February 7, 2017

    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 G∞p of the maximal unramified pro-p extension of K is to determine its two-stage approximation M=G2pk, that is the second derived quotient M≃G/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.