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Algebraic integers in totally imaginary quartic number fields are notdiscrete in the complex plane under a fixed embedding, which makes itimpossible to visualize all integers in the plane, unlike the quadraticimaginary algebraic integers. In this note we consider a naturally occurringdiscrete subset of the algebraic integers with similar properties as lattices.For the fifth cyclotomic field, we investigate those integers with absolutevalues under a fixed embedding in a given bound. We show that such integersform a discrete set in the complex plane. It is observed that this subset hasquasi-periodic appearance. In particular, we also show that the distancebetween a fixed point to the most adjacent point in this subset takes only twopossible values.

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