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28 March 2024
 
  » arxiv » 1601.0179

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Index-p abelianization data of p-class tower groups, II
Daniel C. Mayer ;
Date 2 Jan 2016
AbstractLet (p) be a prime and (K) be a number field with non-trivial (p)-class group (mathrm{Cl}_p(K)). An important step in identifying the Galois group (G=mathrm{G}_p^infty(K)) of the maximal unramified pro-(p) extension of (K) is to determine its two-stage approximation (mathfrak{M}=mathrm{G}_p^2(K)), that is the second derived quotient (mathfrak{M}simeq G/G^{primeprime}). The family of abelian type invariants of the (p)-class groups (mathrm{Cl}_p(L)) of all unramified cyclic extensions (Lvert K) of degree (p) is called the extit{index-(p) abelianization data} (IPAD) ( au_1(K)) of (K) and has turned out to be useful for determining the second (p)-class group (mathfrak{M}). In this paper we introduce two different kinds of extit{generalized} IPADs for obtaining more sophisticated results. The extit{multi-layered} IPAD (( au_1(K), au_2(K))) includes data on unramified abelian extensions (Lvert K) of degree (p^2) and enables sharper bounds for the order of (mathfrak{M}) if (mathrm{Cl}_p(K)simeq (p,p,p)). The extit{iterated} IPAD of extit{second order} ( au^{(2)}(K)) contains information on non-abelian unramified extensions (Lvert K) of degrees (p^2) or even (p^3) and admits the identification of the (p)-class tower group (G) for various series of real quadratic fields (K=mathbb{Q}(sqrt{d})), (d>0), with (mathrm{Cl}_p(K)simeq (p,p)) having a (p)-class field tower of exact length (ell_p(K)=3) as a striking novelty.
Source arXiv, 1601.0179
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