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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 kK 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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References

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Correspondence to Yu. Yu. Kochetkov.

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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

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Keywords

  • Convex Hull
  • Convex Cone
  • Galois Group
  • Fundamental Domain
  • Positive Element