<abstract xmlns="http://www.w3.org/1999/xhtml"><p>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 [<xref ref-type="bibr" rid="j_forma-2016-0025_ref_014_w2aab2b8b6b1b7b1ab1ac14Aa">14</xref>], [<xref ref-type="bibr" rid="j_forma-2016-0025_ref_001_w2aab2b8b6b1b7b1ab1ab1Aa">1</xref>], [<xref ref-type="bibr" rid="j_forma-2016-0025_ref_010_w2aab2b8b6b1b7b1ab1ac10Aa">10</xref>] and [<xref ref-type="bibr" rid="j_forma-2016-0025_ref_007_w2aab2b8b6b1b7b1ab1ab7Aa">7</xref>] 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[<italic>x</italic>] turns to be a field.</p></abstract>

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.

Algebraic Numbers

Formalized Mathematics

This work is licensed under version 3.0 of the Creative Commons Attribution–ShareAlike License.

{"article-title":"Algebraic Numbers"}

This article provides definitions and examples upon an integral element of unital commutative rings. An algebraic number is also treated as consequence of a...

Algebraic Numbers

Online veröffentlicht: 23. Feb. 2017

Seitenbereich: 291 - 299

Eingereicht: 15. Dez. 2016

DOI: https://doi.org/10.1515/forma-2016-0025

Schlüsselwörter
algebraic number, integral dependency

© 2016 Yasushige Watase, published by De Gruyter Open

This work is licensed under version 3.0 of the Creative Commons Attribution–ShareAlike License.

Algebraic Numbers

Online veröffentlicht: 23. Feb. 2017

Seitenbereich: 291 - 299

Eingereicht: 15. Dez. 2016

DOI: https://doi.org/10.1515/forma-2016-0025

Schlüsselwörteralgebraic number, integral dependency

© 2016 Yasushige Watase, published by De Gruyter Open

This work is licensed under version 3.0 of the Creative Commons Attribution–ShareAlike License.

Schlüsselwörter
algebraic number, integral dependency