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The fundamental theorem of algebra for Hamilton and Cayley numbers. (English) Zbl 1144.30004
Authors’ summary: In this paper we prove the fundamental theorem of algebra for polynomials with coefficients in the skew field of Hamilton numbers (quaternions) and in the division algebra of Cayley numbers (octonions). The proof, inspired by recent definitions and results on regular functions of a quaternionic and of a octonionic variable, follows the guidelines of the classical topological argument due to Gauss.

MSC:
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
30G35 Functions of hypercomplex variables and generalized variables
32A30 Other generalizations of function theory of one complex variable (should also be assigned at least one classification number from Section 30-XX)
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[1] Baez J. (2002). The octonions. Bull. Am. Math. Soc. 39: 145–205 · Zbl 1026.17001 · doi:10.1090/S0273-0979-01-00934-X
[2] Bredon G.E. (1993). Topology and Geometry. GTM, vol. 139. Springer, New York · Zbl 0791.55001
[3] do Carmo M.P. (1993). Riemannian Geometry. Birkhäuser, Boston · Zbl 0505.53001
[4] Eilenberg S. and Niven I. (1944). The ”Fundamental Theorem of Algebra” for quaternions; Bull. Am. Math. Soc. 50: 246–248 · Zbl 0063.01228 · doi:10.1090/S0002-9904-1944-08125-1
[5] Gentili G. and Struppa D.C. (2006). A new approach to Cullen-regular functions of a quaternionic variable. C. R. Acad. Sci. Paris I 342: 741–744 · Zbl 1105.30037
[6] Gentili G. and Struppa D.C. (2007). A new theory of regular functions of a quaternionic variable. Adv. Math. 216: 279–301 · Zbl 1124.30015 · doi:10.1016/j.aim.2007.05.010
[7] Gentili, G., Struppa, D.C.: Regular functions on the space of Cayley numbers. Dipartimento di Matematica ”U. Dini”, Università di Firenze, n. 13 (2006, preprint) · Zbl 1105.30037
[8] Hirsch M.W. (1976). Differential Topology. GTM, vol. 33. Springer, New York · Zbl 0356.57001
[9] Niven J. (1941). Equations in quaternions. Am. Math. Mon. 48: 654–661 · Zbl 0060.08002 · doi:10.2307/2303304
[10] Niven J. (1942). The roots of a quaternion. Am. Math. Mon. 49: 386–388 · Zbl 0061.01407 · doi:10.2307/2303134
[11] Pogorui A. and Shapiro M. (2004). On the structure of the Set of Zeros of Quaternionic Polynomials. Complex Var. Theory Appl. 49(6): 379–389 · Zbl 1160.30353
[12] Sudbery A. (1979). Quaternionic analysis. Math. Proc. Camb. Philos. Soc. 85: 199–225 · Zbl 0399.30038 · doi:10.1017/S0305004100055638
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