Hostname: page-component-cb9f654ff-h4f6x Total loading time: 0 Render date: 2025-08-28T22:58:42.033Z Has data issue: false hasContentIssue false
  • English
  • Français

A NOTE ON THE FUNDAMENTAL THEOREM OF ALGEBRA

Part of: General field theory

Published online by Cambridge University Press:  28 March 2018

MOHSEN ALIABADI Open the ORCID record for MOHSEN ALIABADI [Opens in a new window]
MOHSEN ALIABADI*
Affiliation:
Department of Mathematics, Statistics and Computer Science, University of Illinois, 851 S. Morgan St, Chicago, IL 60607, USA email maliab2@uic.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The algebraic proof of the fundamental theorem of algebra uses two facts about real numbers. First, every polynomial with odd degree and real coefficients has a real root. Second, every nonnegative real number has a square root. Shipman [‘Improving the fundamental theorem of algebra’, Math. Intelligencer29(4) (2007), 9–14] showed that the assumption about odd degree polynomials is stronger than necessary; any field in which polynomials of prime degree have roots is algebraically closed. In this paper, we give a simpler proof of this result of Shipman.

Keywords

fundamental theorem of algebrafundamental theorem of Galois theoryperfect field

MSC classification

Secondary: 12E05: Polynomials (irreducibility, etc.)

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 97 , Issue 3 , June 2018 , pp. 382 - 385
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

References

Basu, S. and Velleman, D. J., ‘On Gauss’s first proof of the fundamental theorem of algebra’, Amer. Math. Monthly 124(8) (2017), 688–694.CrossRefGoogle Scholar
Carrell, J. B., Groups, Matrices and Vector Spaces. A Group Theoretic Approach to Linear Algebra (Springer, New York, 2017).Google Scholar
Eisermann, M., ‘The fundamental theorem of algebra made effective: an elementary real-algebraic proof via Sturm chains’, Amer. Math. Monthly 119(9) (2012), 715–752.Google Scholar
Fine, B. and Rosenberger, G., The Fundamental Theorem of Algebra (Springer, New York, 1997).Google Scholar
Levin, A., Difference Algebra, Algebra and Applications, Vol. 8 (Springer, Netherlands, 2008).Google Scholar
Shipman, J., ‘Improving the fundamental theorem of algebra’, Math. Intelligencer 29(4) (2007), 9–14.Google Scholar