{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T00:47:34Z","timestamp":1777423654174,"version":"3.51.4"},"reference-count":14,"publisher":"Walter de Gruyter GmbH","issue":"4","license":[{"start":{"date-parts":[[2016,12,1]],"date-time":"2016-12-01T00:00:00Z","timestamp":1480550400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-sa\/3.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2016,12,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats: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 \u201cintegral\u201d. 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 \u211a induced by substitution of an algebraic number to the polynomial ring of \u211a[<jats:italic>x<\/jats:italic>] turns to be a field.<\/jats:p>","DOI":"10.1515\/forma-2016-0025","type":"journal-article","created":{"date-parts":[[2017,2,25]],"date-time":"2017-02-25T10:00:53Z","timestamp":1488016853000},"page":"291-299","source":"Crossref","is-referenced-by-count":3,"title":["Algebraic Numbers"],"prefix":"10.1515","volume":"24","author":[{"given":"Yasushige","family":"Watase","sequence":"first","affiliation":[{"name":"Suginami-ku Matsunoki 6, 3-21 Tokyo, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2017,2,23]]},"reference":[{"key":"2021040706190683399_j_forma-2016-0025_ref_001_w2aab2b8b6b1b7b1ab1ab1Aa","unstructured":"[1] Michael Francis Atiyah and Ian Grant Macdonald. Introduction to Commutative Algebra, volume 2. Addison-Wesley Reading, 1969."},{"key":"2021040706190683399_j_forma-2016-0025_ref_002_w2aab2b8b6b1b7b1ab1ab2Aa","unstructured":"[2] Jonathan Backer, Piotr Rudnicki, and Christoph Schwarzweller. Ring ideals. Formalized Mathematics, 9(3):565\u2013582, 2001."},{"key":"2021040706190683399_j_forma-2016-0025_ref_003_w2aab2b8b6b1b7b1ab1ab3Aa","unstructured":"[3] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41\u201346, 1990."},{"key":"2021040706190683399_j_forma-2016-0025_ref_004_w2aab2b8b6b1b7b1ab1ab4Aa","unstructured":"[4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107\u2013114, 1990."},{"key":"2021040706190683399_j_forma-2016-0025_ref_005_w2aab2b8b6b1b7b1ab1ab5Aa","unstructured":"[5] Czes\u0142aw Byli\u0144ski. Functions and their basic properties. Formalized Mathematics, 1(1): 55\u201365, 1990."},{"key":"2021040706190683399_j_forma-2016-0025_ref_006_w2aab2b8b6b1b7b1ab1ab6Aa","unstructured":"[6] Eugeniusz Kusak, Wojciech Leo\u0144czuk, and Micha\u0142 Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335\u2013342, 1990."},{"key":"2021040706190683399_j_forma-2016-0025_ref_007_w2aab2b8b6b1b7b1ab1ab7Aa","unstructured":"[7] Hideyuki Matsumura. Commutative Ring Theory. Cambridge University Press, 2nd edition, 1989. Cambridge Studies in Advanced Mathematics."},{"key":"2021040706190683399_j_forma-2016-0025_ref_008_w2aab2b8b6b1b7b1ab1ab8Aa","unstructured":"[8] Robert Milewski. The ring of polynomials. Formalized Mathematics, 9(2):339\u2013346, 2001."},{"key":"2021040706190683399_j_forma-2016-0025_ref_009_w2aab2b8b6b1b7b1ab1ab9Aa","unstructured":"[9] Robert Milewski. The evaluation of polynomials. Formalized Mathematics, 9(2):391\u2013395, 2001."},{"key":"2021040706190683399_j_forma-2016-0025_ref_010_w2aab2b8b6b1b7b1ab1ac10Aa","unstructured":"[10] Masayoshi Nagata. Theory of Commutative Fields, volume 125. American Mathematical Society, 1985. Translations of Mathematical Monographs."},{"key":"2021040706190683399_j_forma-2016-0025_ref_011_w2aab2b8b6b1b7b1ab1ac11Aa","unstructured":"[11] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329\u2013334, 1990."},{"key":"2021040706190683399_j_forma-2016-0025_ref_012_w2aab2b8b6b1b7b1ab1ac12Aa","unstructured":"[12] Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569\u2013573, 1990."},{"key":"2021040706190683399_j_forma-2016-0025_ref_013_w2aab2b8b6b1b7b1ab1ac13Aa","unstructured":"[13] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291\u2013296, 1990."},{"key":"2021040706190683399_j_forma-2016-0025_ref_014_w2aab2b8b6b1b7b1ab1ac14Aa","unstructured":"[14] Oscar Zariski and Pierre Samuel. Commutative Algebra I. Springer, 2nd edition, 1975."}],"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/content.sciendo.com\/view\/journals\/forma\/24\/4\/article-p291.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.sciendo.com\/article\/10.1515\/forma-2016-0025","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,4,7]],"date-time":"2021-04-07T20:26:32Z","timestamp":1617827192000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.sciendo.com\/article\/10.1515\/forma-2016-0025"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,12,1]]},"references-count":14,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2017,2,23]]},"published-print":{"date-parts":[[2016,12,1]]}},"alternative-id":["10.1515\/forma-2016-0025"],"URL":"https:\/\/doi.org\/10.1515\/forma-2016-0025","relation":{},"ISSN":["1898-9934","1426-2630"],"issn-type":[{"value":"1898-9934","type":"electronic"},{"value":"1426-2630","type":"print"}],"subject":[],"published":{"date-parts":[[2016,12,1]]}}}