<abstract xmlns="http://www.w3.org/1999/xhtml"><p>In this article we further extend the algebraic theory of polynomial rings in Mizar [<xref ref-type="bibr" rid="j_forma-2017-0018_ref_001_w2aab3b7b2b1b6b1ab1ab1Aa">1</xref>, <xref ref-type="bibr" rid="j_forma-2017-0018_ref_002_w2aab3b7b2b1b6b1ab1ab2Aa">2</xref>, <xref ref-type="bibr" rid="j_forma-2017-0018_ref_003_w2aab3b7b2b1b6b1ab1ab3Aa">3</xref>]. We deal with roots and multiple roots of polynomials and show that both the real numbers and finite domains are not algebraically closed [<xref ref-type="bibr" rid="j_forma-2017-0018_ref_005_w2aab3b7b2b1b6b1ab1ab5Aa">5</xref>, <xref ref-type="bibr" rid="j_forma-2017-0018_ref_007_w2aab3b7b2b1b6b1ab1ab7Aa">7</xref>]. We also prove the identity theorem for polynomials and that the number of multiple roots is bounded by the polynomial’s degree [<xref ref-type="bibr" rid="j_forma-2017-0018_ref_004_w2aab3b7b2b1b6b1ab1ab4Aa">4</xref>, <xref ref-type="bibr" rid="j_forma-2017-0018_ref_006_w2aab3b7b2b1b6b1ab1ab6Aa">6</xref>].</p></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].

On Roots of Polynomials and Algebraically Closed Fields

Formalized Mathematics

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"On Roots of Polynomials and Algebraically Closed Fields"}

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...

On Roots of Polynomials and Algebraically Closed Fields

Published Online: Dec 19, 2017

Page range: 185 - 195

Received: Aug 30, 2017

DOI: https://doi.org/10.1515/forma-2017-0018

Keywords
commutative algebra, polynomials, algebraic closed fields

© 2017 Christoph Schwarzweller, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

On Roots of Polynomials and Algebraically Closed Fields

Published Online: Dec 19, 2017

Page range: 185 - 195

Received: Aug 30, 2017

DOI: https://doi.org/10.1515/forma-2017-0018

Keywordscommutative algebra, polynomials, algebraic closed fields

© 2017 Christoph Schwarzweller, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
commutative algebra, polynomials, algebraic closed fields