1. Cluster Magnification, Root Capacity, Unique Chains and Base Change
- Author
-
Bhagwat, Chandrasheel and Jaiswal, Shubham
- Subjects
Mathematics - Group Theory ,Mathematics - Number Theory ,11R32, 12F05, 12F10 - 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. Finally we establish results about all these phenomena under a particular type of base change. The article concludes with results about strong cluster magnification and unique chains and some properties of the ascending index for a field extension., Comment: 34 pages. Changes in v3 : Significant changes in old sections of v2. New section on ascending index added to v2. Changes in v2 : Changed the title (previous title was "Inverse Cluster Magnifi..."). Significant changes in old sections of v1. New sections on root capacity, unique chains and base change added to v1. Submitted to Journal of Number Theory
- Published
- 2024