Periodic sequences of p-class tower groups

TL;DR

This paper demonstrates that p-class tower groups of certain multiquadratic fields form periodic sequences in descendant trees, with their identification reducible to simple computations of low-degree invariants.

Contribution

It reveals the periodic structure of p-class tower groups in descendant trees and links their determination to easily computable arithmetical invariants.

Findings

p-class tower groups form periodic sequences in descendant trees

The specific tower group is determined by the p-class number of an auxiliary field

Identification of the tower group reduces to low-degree invariant computations

Abstract

Recent examples of periodic bifurcations in descendant trees of finite p-groups with p in {2,3} are used to show that the possible p-class tower groups G of certain multiquadratic fields K with p-class group of type (2,2,2), resp. (3,3), form periodic sequences in the descendant tree of the elementary abelian root C(2)xC(2)xC(2), resp. C(3)xC(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.