A Purely Algebraic Proof of the Fundamental Theorem of Algebra
Keywords:
fundamental theorem of algebra, continuity, real closed field, intermediate value theorem, ultrapowerAbstract
Proofs of the fundamental theorem of algebra can be divided upinto three groups according to the techniques involved: proofs that rely onreal or complex analysis, algebraic proofs, and topological proofs. Algebraicproofs make use of the fact that odd-degree real polynomials have real roots.This assumption, however, requires analytic methods, namely, the intermediatevalue theorem for real continuous functions. In this paper, we developthe idea of algebraic proof further towards a purely algebraic proof of theintermediate value theorem for real polynomials. In our proof, we neither usethe notion of continuous function nor refer to any theorem of real and complexanalysis. Instead, we apply techniques of modern algebra: we extend thefield of real numbers to the non-Archimedean field of hyperreals via an ultraproductconstruction and explore some relationships between the subringof limited hyperreals, its maximal ideal of infinitesimals, and real numbers.Downloads
Metrics
References
Artin, E., Schreier, O.: 2007, Algebraische Konstruktion reeller Körper, Abhandlungen aus dem Mathematischen Seminar der Hamburgischen Universität 5 (1926), 85–99; The algebraic construction of real fields, in: M. Rosen (ed.), Exposition by Emil Artin, AMS-LMS, 107–118.
Bair, J., et al.: 2013, Is mathematical history written by the victors?, Notices of The American Mathematical Society 7, 886–904.
Błaszczyk, P.: 2007, Analiza filozoficzna rozprawy Richarda Dedekinda Stetigkeit und irrationale Zahlen, Wydawnictwo Naukowe AP w Krakowie, Kraków.
Birkhoff, G., Mac Lane, S.: 1977, A Survey of Modern Algebra, Macmillan Publishing, New York.
Bolzano, B.: 2004, Purely analytic proof of the theorem that between any two values which give results of opposite signs, there lies at least one real root of the equation, in: S. Russ (ed.), The Mathematical Works of Bernard Bolzano, Oxford University Press, Oxford, 251–263.
Cantor, G.: 1895, Beiträge zur Begründung der transfiniten Mengenlehre, Mathematische Annalen 46, 481–512.
Cohen, L. W., Ehrlich, G.: 1963, The Structure of the Real Number System, Van Nostrand Co., Princeton, New Jersey.
Cohen, P. M.: 1991, Algebra, Vol. III, John Wiley & Sons, Chichester.
Dedekind, R.: 1872, Stetigkeit und irrationale Zahlen, Friedr. Vieweg & Sohn, Braunschweig.
Derksen, H.: 2003, The fundamental theorem of algebra and linear algebra, The American Mathematical Monthly 11, 620–623.
Fine, B., Rosenberger, B.: 1997, The Fundamental Theorem of Algebra, Springer, New York.
Goldblatt, R.: 1998, Lectures on the Hyperreals, Springer, New York.
Hall, J. F.: 2011, Completeness of ordered fields, http://arxiv.org/abs/1101.5652v1.
Jacobson, N.: 1975, Lectures in Abstract Algebra, Vol. III, Van Nostrand Co., Princeton, New Jersey.
Kuratowski, K., Mostowski, A.: 1966, Set Theory, North-Holland, PWN, Amsterdam, Warsaw.
Libman, G.: 2006, A Nonstandard Proof of the Fundamental Theorem of Algebra, The American Mathematical Monthly 4, 347–348.
Maor, E.: 2007, The Pythagorean Theorem: A 4,000-Year History, Princeton University Press, Princeton, New Jersey.
Marker, D.: 2002, Model Theory. An introduction, Springer, New York.
Łoś, J.: 1955, Quelques remarques, théorèmes et problèmes sur les classes définissables d’algèbres, in: T. Skolem et al. (ed.), Mathematical interpretation of formal systems, North-Holland, Amsterdam, 98–113.
Remmert, R.: 1995, The Fundamental Theorem of Algebra, in: H. D. Ebbinghaus et al. (ed.), Numbers, Springer, New York, 97–122.
Riemenschneider, O.: 2001, 37 elementare axiomatische Charakterisierungen des reellen Zahlkörpers, Mittelungen der Mathematischen Gesellschaft in Hamburg 20, 71–95.
Teismann, H.: 2013, Toward a More Complete List of Completeness Axioms, The American Mathematical Monthly 2, 99–114.
The Bibliography for the Fundamental Theorem of Algebra.: http://mathfaculty.fullerton.edu/mathews/c2003/FunTheoremAlgebraBib/ Links/FunTheoremAlgebraBib_lnk_3.html.
Zassenhaus, H.: 1967, On the Fundamental Theorem of Algebra, The American Mathematical Monthly 74, 485–497.
Downloads
Published
How to Cite
Issue
Section
License
Read the full statement of the license to publish PDF file.
(see English version) (see Polish version)
