Geometry of totally real Galois fields of degree 4

TL;DR

This paper studies the geometric structure of totally real Galois fields of degree 4, focusing on the convex cones of positive elements and the action of units on their boundary polyhedrons, providing specific examples.

Contribution

It introduces the geometric framework for analyzing units in degree 4 Galois fields and explores fundamental domains of their action on boundary polyhedrons.

Findings

Description of the convex cone of positive elements

Analysis of the boundary polyhedrons and their structure

Examples of fundamental domains for the unit group action

Abstract

We will consider a totally real Galois field $K$ of degree 4 as the linear coordinate space $\mathbb{Q}^4\subset\mathbb{R}^4$. An element $k\in K$ is called strictly positive, if all its conjugates are positive. The set of strictly positive elements is a convex cone in $K$. The convex hull of strictly positive integral elements is a convex subset of this cone and its boundary $\Gamma$ is an infinite union of 3-dimensional polyhedrons. The group $U$ of strictly positive units acts on $\Gamma$: the action of a strictly positive unit permutes polyhedrons. Fundamental domains of this action are the object of study in this work. We mainly present some interesting examples.