Algebraic entanglement arises when replacing the usual second-order symmetry between positive and negative real numbers by a symmetry of order three related to the definition of the identity element for addition of numbers having this symmetry. With this notion, we form a field  where the additive associativity is assisted (or aaa), meaning that it is slightly more demanding than the current one. This aaa-field contains three distinct entangled copies of R. The Cayley-Dickson doubling procedure applied three times successively to  gives three sets of numbers  and  which are division and quadratically normed aaa-algebras of  and  entangled real dimensions, respectively. The multiplication of  is noncommutative, and that of  is nonassociative. Basic mathematical analysis on  shows that differentiability of a function at a point does not imply its continuity at that point. Within a geometric representation of  in  we also find key analogies between its basic mathematical analysis and that on C, including an analog of the Cauchy integral formula.

algebraic structures, algebraic entanglement, fields with sum of squares, compound additive inverse, assisted additive associativity, integrals of Cauchy type.

Received: April 10, 2024; Accepted: June 5, 2024; Published: July 13, 2024

How to cite this article: Claude Gauthier, On algebraic entangled algebras and fields, JP Journal of Algebra, Number Theory and Applications 63(5) (2024), 413-434. https://doi.org/10.17654/0972555524025

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:
[1] F. G. Frobenius, Über lineare substitutionen und bilineare Formen, Journal für die reine und angewandte Mathematik 84 (1878), 1-63.
[2] H. Hopf, Ein topologischer Beitrag zur reellen algebra, Comment. Math. Helv. 13 (1940), 219-239.
[3] M. Kervaire, Non-parallelizability of the n-sphere for Proc. Nat. Acad. Sci. 44 (1958), 280-283.
[4] R. Bott and J. Milnor, On the parallelizability of spheres, Bull. A.M.S. 64 (1958), 87-89.
[5] A. Hurwitz, Über die composition der quadratischen Formen von beliebig vielen Variabeln, Nachr. Ges. Wiss. Gottingen (1898), 309-316.
[6] T.-Y. Lam, Introduction to Quadratic Forms over Fields, A.M.S., Providence, 2005.
[7] C. Gauthier, On tridemi-real number system and algebraic entanglement, JP Journal of Algebra, Number Theory and Applications 56 (2022), 71-94.
[8] I. L. Kantor and A. S. Solodovnikov, Hypercomplex Numbers, Springer-Verlag, New York, 1989.
[9] R. V. Churchill and J. W. Brown, Complex Variables and Applications, Fourth Edition, McGraw-Hill, New York, 1984.