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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

4 Issues per year


SCImago Journal Rank (SJR) 2016: 0.207
Source Normalized Impact per Paper (SNIP) 2016: 0.315

Open Access
Online
ISSN
1898-9934
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Volume 24, Issue 4 (Dec 2016)

Issues

Algebraic Numbers

Yasushige Watase
Published Online: 2017-02-23 | DOI: https://doi.org/10.1515/forma-2016-0025

Summary

This article provides definitions and examples upon an integral element of unital commutative rings. An algebraic number is also treated as consequence of a concept of “integral”. Definitions for an integral closure, an algebraic integer and a transcendental numbers [14], [1], [10] and [7] are included as well. As an application of an algebraic number, this article includes a formal proof of a ring extension of rational number field ℚ induced by substitution of an algebraic number to the polynomial ring of ℚ[x] turns to be a field.

Keywords: algebraic number; integral dependency

MSC 2010: 11R04; 13B21; 03B35

References

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  • [2] Jonathan Backer, Piotr Rudnicki, and Christoph Schwarzweller. Ring ideals. Formalized Mathematics, 9(3):565–582, 2001.Google Scholar

  • [3] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.Google Scholar

  • [4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.Google Scholar

  • [5] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Google Scholar

  • [6] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.Google Scholar

  • [7] Hideyuki Matsumura. Commutative Ring Theory. Cambridge University Press, 2nd edition, 1989. Cambridge Studies in Advanced Mathematics. Google Scholar

  • [8] Robert Milewski. The ring of polynomials. Formalized Mathematics, 9(2):339–346, 2001.Google Scholar

  • [9] Robert Milewski. The evaluation of polynomials. Formalized Mathematics, 9(2):391–395, 2001.Google Scholar

  • [10] Masayoshi Nagata. Theory of Commutative Fields, volume 125. American Mathematical Society, 1985. Translations of Mathematical Monographs. Google Scholar

  • [11] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329–334, 1990.Google Scholar

  • [12] Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569–573, 1990.Google Scholar

  • [13] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.Google Scholar

  • [14] Oscar Zariski and Pierre Samuel. Commutative Algebra I. Springer, 2nd edition, 1975. Google Scholar

About the article

Received: 2016-12-15

Published Online: 2017-02-23

Published in Print: 2016-12-01


Citation Information: Formalized Mathematics, ISSN (Online) 1898-9934, ISSN (Print) 1426-2630, DOI: https://doi.org/10.1515/forma-2016-0025.

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© 2016 Yasushige Watase, published by De Gruyter Open. This work is licensed under version 3.0 of the Creative Commons Attribution–ShareAlike License. BY-SA 3.0

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