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arXiv · 1209.0412

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

Wenhan Wang
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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.

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Wenhan Wang. 2012-09-03. Discrete Subsets of Totally Imaginary Quartic Algebraic Integers in the Complex Plane. https://arxiv.org/abs/1209.0412

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