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STRICTLY REAL FUNDAMENTAL THEOREM OF ALGEBRA USING POLYNOMIAL INTERLACING

Part of: Polynomials, rational functions Real and complex fields

Published online by Cambridge University Press:  18 January 2021

SOHAM BASU Open the ORCID record for SOHAM BASU [Opens in a new window]
SOHAM BASU*
Affiliation:
School of Engineering, École Polytechnique Fédérale de Lausanne, Lausanne1015, Switzerland
*
e-mail: [email protected]
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Abstract

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Without resorting to complex numbers or any advanced topological arguments, we show that any real polynomial of degree greater than two always has a real quadratic polynomial factor, which is equivalent to the fundamental theorem of algebra. The proof uses interlacing of bivariate polynomials similar to Gauss’s first proof of the fundamental theorem of algebra using complex numbers, but in a different context of division residues of strictly real polynomials. This shows the sufficiency of basic real analysis as the minimal platform to prove the fundamental theorem of algebra.

Keywords

fundamental theorem of algebrapolynomial interlacing

MSC classification

Primary: 12D05: Polynomials
Secondary: 12D10: Polynomials 26C10: Polynomials
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 104 , Issue 2 , October 2021 , pp. 249 - 255
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© Australian Mathematical Publishing Association Inc. 2021.

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