On Variants of Inverse Cluster Size Problem & General Magnification

TL;DR

This paper explores variants of the inverse cluster size problem in number theory, introducing primitive extensions and general magnification concepts, and provides solutions for specific cases including totally real number fields.

Contribution

It introduces the primitive variant of the inverse cluster size problem and establishes existence results for primitive extensions over number fields of any degree.

Findings

Existence of primitive extensions over number fields of any degree

Complete resolution of the totally real variant of the problem

Introduction of strong and weak general magnification concepts

Abstract

In this article we establish certain variants of the Inverse Cluster Size problem. We introduce the notion of primitive extensions and establish the Primitive variant of the problem. Precisely, we prove the existence of primitive extensions over number fields of any given degree and cluster size less than the degree. We also introduce the notions of Strong and Weak General Magnification and the notion of general primitive extensions. We establish some interesting cases of the General primitive variant of the problem. We also recall the notion of totally real number fields and resolve the Totally real variant of the problem completely.