On Roots of Polynomials and Algebraically Closed Fields

Christoph Schwarzweller

doi:10.1515/forma-2017-0018

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On Roots of Polynomials and Algebraically Closed Fields

Formalized Mathematics

Volume 25 (2017): Issue 3 (October 2017)

By: Christoph Schwarzweller

Open Access

|Dec 2017

Abstract

In this article we further extend the algebraic theory of polynomial rings in Mizar [1, 2, 3]. We deal with roots and multiple roots of polynomials and show that both the real numbers and finite domains are not algebraically closed [5, 7]. We also prove the identity theorem for polynomials and that the number of multiple roots is bounded by the polynomial’s degree [4, 6].

References

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[3] Adam Grabowski, Artur Korniłowicz, and Christoph Schwarzweller. On algebraic hierarchies in mathematical repository of Mizar. In M. Ganzha, L. Maciaszek, and M. Paprzycki, editors, Proceedings of the 2016 Federated Conference on Computer Science and Information Systems (FedCSIS), volume 8 of Annals of Computer Science and Information Systems, pages 363–371, 2016. doi:10.15439/2016F520.10.15439/2016F520
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[5] Nathan Jacobson. Basic Algebra I. 2nd edition. Dover Publications Inc., 2009.
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[7] Knut Radbruch. Algebra I. Lecture Notes, University of Kaiserslautern, Germany, 1991.
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[8] Christoph Schwarzweller and Agnieszka Rowińska-Schwarzweller. Schur’s theorem on the stability of networks. Formalized Mathematics, 14(4):135–142, 2006. doi:10.2478/v10037-006-0017-9.10.2478/v10037-006-0017-9
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[9] Christoph Schwarzweller, Artur Korniłowicz, and Agnieszka Rowińska-Schwarzweller. Some algebraic properties of polynomial rings. Formalized Mathematics, 24(3):227–237, 2016. doi:10.1515/forma-2016-0019.10.1515/forma-2016-0019
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Articles in this issue

DOI: https://doi.org/10.1515/forma-2017-0018 | Journal eISSN: 1898-9934 | Journal ISSN: 1426-2630

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Language: English

Page range: 185 - 195

Submitted on: Aug 30, 2017

Published on: Dec 19, 2017

Published by: University of Białystok

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

commutative algebra,

polynomials,

algebraic closed fields

Related subjects:

Mathematics,

General mathematics,

Computer sciences,

Computer sciences, other

© 2017 Christoph Schwarzweller, published by University of Białystok
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Volume 25 (2017): Issue 3 (October 2017)