Geometry of Totally Real Galois Fields of Degree 4

Abstract

We consider a totally real Galois field K of degree 4 as the linear coordinate space ℚ⁴ ⊂ ℝ⁴. An element k ∈ K is called strictly positive if all of its conjugates are positive. The set of strictly positive elements is a convex cone in ℚ⁴. The convex hull of strictly positive integral elements is a convex subset of this cone and its boundary Γ is an infinite union of 3-dimensional polyhedrons. The group U of strictly positive units acts on Γ: the action of a strictly positive unit permutes polyhedrons. Examples of fundamental domains of this action are the object of study in this work.