Periodic Sequences of p-Class Tower Groups

Abstract

Recent examples of periodic bifurcations in descendant trees of finite p-groups with p {2, 3}∈ are used to show that the possible p-class tower groups G of certain multiquadratic fields K with pclass group of type (2, 2, 2) , resp. (3, 3) , form periodic sequences in the descendant tree of the elementary Abelian root C 3 2 , resp. C 2 3 . The particular vertex of the periodic sequence which occurs as the p-class tower group G of an assigned field K is determined uniquely by the p-class number of a quadratic, resp. cubic, auxiliary field k, associated unambiguously to K. Consequently, the hard problem of identifying the p-class tower group G is reduced to an easy computation of low degree arithmetical invariants.

Artin Transfer Patterns on Descendant Trees of Finite p-Groups

“…for selecting among several candidates G for the p-class tower group, in dependence on the torsion-free unit rank of the base field K, for instance in [21] [23].…”

New number fields with known p-class tower

“…We discovered that the corresponding 3-class tower groups G = Gal F ∞ 3 (k)|k with G/G G are sporadic isolated vertices (outside of coclass trees) with unbounded higher coclass cc(G) ≥ 3 but with fixed derived length dl(G) = 3, as visualized in [21,. These results concerning the section E were generalized in [22,.…”

Criteria for Three-Stage Towers of <i>p</i>-Class Fields

“…Then we generally put 3 p = and consider 3-class tower groups ( §7). In §5, we first restate a summary of all possible IPADs of a number field K with 3-class group 3 Cl K of type (3,3) [ [1], Thm. 3.1-3.2, pp.…”

“…Cl K of type (3,3). In §7.2, results for the dominant scenario with 3-principalization 1 K  of type a are given.…”

“…2 K τ contains information on non-abelian unramified extensions L/K of degree p 2 , or even p 3 , and admits the identification of the p-class tower group G for various infinite series of quadratic fields ( )…”