Skip to main content
SpringerLink
Log in

The fundamental theorem of algebra for Hamilton and Cayley numbers

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Baez J. (2002). The octonions. Bull. Am. Math. Soc. 39: 145–205

    Article  MATH  MathSciNet  Google Scholar 

  2. Bredon G.E. (1993). Topology and Geometry. GTM, vol. 139. Springer, New York

    Google Scholar 

  3. do Carmo M.P. (1993). Riemannian Geometry. Birkhäuser, Boston

    Google Scholar 

  4. Eilenberg S. and Niven I. (1944). The “Fundamental Theorem of Algebra” for quaternions; Bull. Am. Math. Soc. 50: 246–248

    Article  MATH  MathSciNet  Google Scholar 

  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

    MATH  MathSciNet  Google Scholar 

  6. Gentili G. and Struppa D.C. (2007). A new theory of regular functions of a quaternionic variable. Adv. Math. 216: 279–301

    Article  MATH  MathSciNet  Google Scholar 

  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)

  8. Hirsch M.W. (1976). Differential Topology. GTM, vol. 33. Springer, New York

    Google Scholar 

  9. Niven J. (1941). Equations in quaternions. Am. Math. Mon. 48: 654–661

    Article  MATH  MathSciNet  Google Scholar 

  10. Niven J. (1942). The roots of a quaternion. Am. Math. Mon. 49: 386–388

    Article  MATH  MathSciNet  Google Scholar 

  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

    MATH  MathSciNet  Google Scholar 

  12. Sudbery A. (1979). Quaternionic analysis. Math. Proc. Camb. Philos. Soc. 85: 199–225

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica “U. Dini”, Università di Firenze, Viale Morgagni 67/A, 50134, Firenze, Italy

    Graziano Gentili & Fabio Vlacci

  2. Department of Mathematics and Computer Sciences, Chapman University, Orange, CA, 92866, USA

    Daniele C. Struppa

Authors
  1. Graziano Gentili
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daniele C. Struppa
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fabio Vlacci
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Graziano Gentili.

Additional information

G. Gentili and F. Vlacci are partially supported by G.N.S.A.G.A. of the I.N.D.A.M. and by M.I.U.R.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gentili, G., Struppa, D.C. & Vlacci, F. The fundamental theorem of algebra for Hamilton and Cayley numbers. Math. Z. 259, 895–902 (2008). https://doi.org/10.1007/s00209-007-0254-9

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-007-0254-9

Mathematics Subject Classification (2000)

Search

Navigation