Cluster Magnification, Root Capacity, Unique Chains and Base Change

TL;DR

This paper explores advanced algebraic concepts like cluster magnification, root capacity, and unique chains, establishing new theoretical results and properties in the context of field extensions and polynomial existence.

Contribution

It generalizes existing results on cluster magnification, introduces new concepts like root capacity and unique chains, and analyzes their properties under base change.

Findings

Existence of polynomials for given degree and cluster size over number fields.

Properties of weak and strong cluster magnification.

Explicit examples of cluster towers and ascending index results.

Abstract

This article is inspired from the work of M Krithika and P Vanchinathan on Cluster Magnification and the work of Alexander Perlis on Cluster Size. We establish the existence of polynomials for given degree and cluster size over number fields which generalises a result of Perlis. We state the Strong cluster magnification problem and establish an equivalent criterion for that. We also discuss the notion of weak cluster magnification and prove some properties. We provide an important example answering a question about Cluster Towers. We introduce the concept of Root capacity and prove some of its properties. We also introduce the concept of unique descending and ascending chains for extensions and establish some properties and explicitly compute some interesting examples. We establish results about all these phenomena under a particular type of base change and discuss some other related results about strong cluster magnification and unique chains. The article concludes with results about ascending index for a field extension which are analogous to results about cluster size.