Searching for Rigidity in Algebraic Starscapes

TL;DR

This paper visualizes algebraic integers in the complex plane, focusing on those with non-full symmetric Galois groups, called rigid, and analyzes their geometric distribution for future research insights.

Contribution

It introduces a novel visualization method highlighting rigid algebraic integers based on Galois invariants, revealing their geometric patterns.

Findings

Rigid algebraic integers form distinct geometric patterns.

Visualization emphasizes non-full symmetric Galois group integers.

Suggests new directions for understanding algebraic integer geometry.

Abstract

We create plots of algebraic integers in the complex plane, exploring the effect of sizing the integers according to various arithmetic invariants. We focus on Galois theoretic invariants, in particular creating plots which emphasize algebraic integers whose Galois group is not the full symmetric group--these integers we call rigid. We then give some analysis of the resulting images, suggesting avenues for future research about the geometry of so-called rigid algebraic integers.