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.
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Yu. Kochetkov, “On geometry of cubic Galois fields,” Math. Notes, 89, 150–155 (2011).
E. B. Vinberg, private communication.
Z. Borevich and I. Shafarevich, Number Theory [in Russian], Nauka, Moscow (1985).
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 19, No. 1, pp. 33–44, 2014.
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Kochetkov, Y.Y. Geometry of Totally Real Galois Fields of Degree 4. J Math Sci 211, 319–326 (2015). https://doi.org/10.1007/s10958-015-2608-x
- Convex Hull
- Convex Cone
- Galois Group
- Fundamental Domain
- Positive Element