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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

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Volume 24, Issue 4


Algebraic Numbers

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


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


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About the article

Received: 2016-12-15

Published Online: 2017-02-23

Published in Print: 2016-12-01

Citation Information: Formalized Mathematics, Volume 24, Issue 4, Pages 291–299, 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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