Lower Bounds for Regulators of Number Fields in terms of their Discriminants

TL;DR

This paper establishes new inequalities linking the regulator of a number field to its discriminant, refining previous results and utilizing recent bounds on heights of relative units to advance understanding in algebraic number theory.

Contribution

It introduces improved inequalities relating regulators and discriminants, building on Silverman's earlier work and incorporating recent bounds on heights of relative units.

Findings

Derived new inequalities connecting regulators and discriminants.

Refined previous bounds on the product of heights of relative units.

Enhanced theoretical understanding of number field invariants.

Abstract

We prove inequalities that compare the regulator of a number field with its absolute discriminant. We refine some ideas in Silverman's work in 1984 where such general inequalities are first proven. In order to prove our main theorems, we combine these refinements with the authors' recent results on bounding the product of heights of relative units in a number field extension.