Index-<i>p</i> Abelianization Data of <i>p</i>-Class Tower Groups

Abstract

Given a fixed prime number p, the multiplet of abelian type invariants of the p-class groups of all unramified cyclic degree p extensions of a number field K is called its IPAD (index-p abelianization data). These invariants have proved to be a valuable information for determining the Galois group p G 2 of the second Hilbert p-class field and the p-capitulation type  of K. For p 3 = and a number field K with elementary p-class group of rank two, all possible IPADs are given in the complete form of several infinite sequences. Iterated IPADs of second order are used to identify the group p G ∞ of the maximal unramified pro-p extension of K.

Periodic Sequences of p-Class Tower Groups

“…At the beginning of 2014, we investigated the root 729, 45 〈 〉, which possesses an infinite balanced cover [2] (Dfn.6.1), and found periodic bifurcations in its decendant tree [2] (Thm.6.4).…”

“…Firstly, since they concern four TKTs E.6, E.14, E.8, E.9 instead of just TKT E.9 [2] (2,2,2) . To appear in J. Algebra Appl.…”

“… is a permutation with five fixed points and a single 2-cycle, and that their first layer TTT 1 τ contains the unique polarized (i.e. parameter dependent) component ( 1,2) m + . Note that the mainline vertices of the tree ( 32,31 ) 〈 〉  reveal the same six stable (i.e.…”

“…167-168). Or they arise from infinite paths of directed edges in descendant trees whose vertices reveal periodic bifurcations (see [1], Thm.21.1, p. 182, [1], Thm.21.3, p. 185, and [2], Thm.6.4). Extensive finite parts of the latter have been constructed computationally with the aid of the p-group generation algorithm by Newman [3] and O'Brien [4] (see [1] [12]- [18]), but their indefinitely repeating periodic pattern has not been proven rigorously, so far.…”

“…[182][183][184][185][186][187][188][189][190][191][192][193] and [2] ( §6.2.2), and summarized in section §2.…”