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Index-p abelianization data of p-class tower groups, II | Daniel C. Mayer
; | Date: |
2 Jan 2016 | Abstract: | Let (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 | Services: | Forum | Review | PDF | Favorites |
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