A Three Dimensional Modification of the Gaussian Number Field

Haluška, Ján; Jastrzębska, Małgorzata

doi:10.2478/tmmp-2019-0020

Abstract

For vectors in E₃ we introduce an associative, commutative and distributive multiplication. We describe the related algebraic and geometrical properties, and hint some applications.

Based on properties of hyperbolic (Clifford) complex numbers, we prove that the resulting algebra 𝕋 is an associative algebra over a field and contains a subring isomorphic to hyperbolic complex numbers. Moreover, the algebra 𝕋 is isomorphic to direct product ℂ×ℝ, and so it contains a subalgebra isomorphic to the Gaussian complex plane.

References

[1] ALPAY, A.—VAJIAC, A.—VAJIAC, M. B.: Gleason,s problem associated to a real ternary algebra and applications, Adv. Appl. Clifford Algebr. 28 (2018), no. 2, pp. 16. (Published online, paper no. 43)10.1007/s00006-018-0857-8
Search in Google Scholar Back to article
[2] DROZD, YU. A.—KIRICHENKO, V. V.: Finite Dimensional Algebras. Springer-Verlag, Berlin, 1994.10.1007/978-3-642-76244-4
Search in Google Scholar Back to article
[3] GREGOR, T.—HALUŠKA, J.: Lexicographical ordering and field operations in the complex plane, Stud. Mat. 41 (2014), no. 2, 123–133.
Search in Google Scholar Back to article
[4] HALUŠKA, J.: On fields inspired with a polar HSV-RGB theory of colour. In: Modern Real Analysis (J. Hejduk, S. Kowalczyk, R. Pawlak, M.Turowska, ed.),Wydawnictwo Universytetu Łódźkiego, Łódź 2015, pp. 69–88.10.18778/7969-663-5.06
Search in Google Scholar Back to article
[5] HARKIN, A. A.—HARKIN, J. B.: Geometry of general complex numbers,Mathematics magazine, 77 (2004), no. 2, 118–129; https://www.researchgate.net/publication/265769569_Geometry_of_Generalized_complex_Numbers10.1080/0025570X.2004.11953236
Search in Google Scholar Back to article
[6] LIPATOV L. N.—RAUSCH DE TRAUTENBERG, M.—VOLKOV G. G.: On the ternary complex analysis and its application, J. Math. Phys. 49 (2008), no. 1, 013502. (Published online) https://doi.org/10.1063/1.282746910.1063/1.2827469
Open DOI Search in Google Scholar Back to article
[7] NECHAEV, V. I.: Number Systems. Izdat. Prosveshchenie, Moscow, 1975. (In Russian)
Search in Google Scholar Back to article
[8] RUDIN, W.: Functional Analysis (2nd edition). In: McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York 1973; In: Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991.
Search in Google Scholar Back to article
[9] PIERCE, R. S.: Associative Algebras. Springer-Verlag, Berlin, 1982.10.1007/978-1-4757-0163-0
Search in Google Scholar Back to article
[10] SNYDER, H. H.: An introduction to theories of regular functions on linear associative mathematics, In: Commutative Algebra. Analytic Methods (Fairfax, Va., 1979), pp. 75–94; In: Lecture Notes in Pure and App. Math. Vol. 68, Dekker, New York, 1982.
Search in Google Scholar Back to article
[11] WARNER, S.: Topological Fields.In: Mathematical Studies Vol. 157.In: Notas Mathematica Vol. 126 (L. Nachbin, ed.), Nord-Holland-Amsterdam-N.Y.-Oxford-Tokyo, 1989.
Search in Google Scholar Back to article
[12] arxiv 1901.08448
Search in Google Scholar Back to article

A Three Dimensional Modification of the Gaussian Number Field

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