Discrete Subsets of Totally Imaginary Quartic Algebraic Integers in the Complex Plane

TL;DR

This paper investigates a specific discrete subset of algebraic integers in totally imaginary quartic fields, demonstrating its discreteness and quasi-periodic structure in the complex plane, with particular focus on the fifth cyclotomic field.

Contribution

It introduces a naturally occurring discrete subset of algebraic integers in totally imaginary quartic fields and analyzes its geometric and structural properties.

Findings

The subset is discrete in the complex plane.

The subset exhibits quasi-periodic appearance.

Distances to the nearest points take only two values.

Abstract

Algebraic integers in totally imaginary quartic number fields are not discrete in the complex plane under a fixed embedding, which makes it impossible to visualize all integers in the plane, unlike the quadratic imaginary algebraic integers. In this note we consider a naturally occurring discrete subset of the algebraic integers with similar properties as lattices. For the fifth cyclotomic field, we investigate those integers with absolute values under a fixed embedding in a given bound. We show that such integers form a discrete set in the complex plane. It is observed that this subset has quasi-periodic appearance. In particular, we also show that the distance between a fixed point to the most adjacent point in this subset takes only two possible values.