{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T01:39:57Z","timestamp":1777426797455,"version":"3.51.4"},"reference-count":9,"publisher":"Walter de Gruyter GmbH","issue":"3","license":[{"start":{"date-parts":[[2017,10,1]],"date-time":"2017-10-01T00:00:00Z","timestamp":1506816000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/3.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2017,10,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p>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\u2019s degree [4, 6].<\/jats:p>","DOI":"10.1515\/forma-2017-0018","type":"journal-article","created":{"date-parts":[[2017,12,20]],"date-time":"2017-12-20T22:16:12Z","timestamp":1513808172000},"page":"185-195","source":"Crossref","is-referenced-by-count":6,"title":["On Roots of Polynomials and Algebraically Closed Fields"],"prefix":"10.1515","volume":"25","author":[{"given":"Christoph","family":"Schwarzweller","sequence":"first","affiliation":[{"name":"Institute of Informatics , University of Gda\u0144sk , Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2017,12,19]]},"reference":[{"key":"2021040808485936510_j_forma-2017-0018_ref_001_w2aab3b7b2b1b6b1ab1ab1Aa","unstructured":"[1] Grzegorz Bancerek, Czes\u0142aw Byli\u0144ski, Adam Grabowski, Artur Korni\u0142owicz, Roman Matuszewski, Adam Naumowicz, Karol P\u0105k, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261\u2013279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007\/978-3-319-20615-8_17.10.1007\/978-3-319-20615-8_17"},{"key":"2021040808485936510_j_forma-2017-0018_ref_002_w2aab3b7b2b1b6b1ab1ab2Aa","unstructured":"[2] Adam Grabowski, Artur Korni\u0142owicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191\u2013198, 2015. doi:10.1007\/s10817-015-9345-1.10.1007\/s10817-015-9345-1"},{"key":"2021040808485936510_j_forma-2017-0018_ref_003_w2aab3b7b2b1b6b1ab1ab3Aa","doi-asserted-by":"crossref","unstructured":"[3] Adam Grabowski, Artur Korni\u0142owicz, and Christoph Schwarzweller. On algebraic hierarchies in mathematical repository of Mizar. In M. Ganzha, L. Maciaszek, and M. Paprzycki, editors, Proceedings of the 2016 Federated Conference on Computer Science and Information Systems (FedCSIS), volume 8 of Annals of Computer Science and Information Systems, pages 363\u2013371, 2016. doi:10.15439\/2016F520.","DOI":"10.15439\/2016F520"},{"key":"2021040808485936510_j_forma-2017-0018_ref_004_w2aab3b7b2b1b6b1ab1ab4Aa","doi-asserted-by":"crossref","unstructured":"[4] H. Heuser. Lehrbuch der Analysis. B.G. Teubner Stuttgart, 1990.","DOI":"10.1007\/978-3-663-12214-2"},{"key":"2021040808485936510_j_forma-2017-0018_ref_005_w2aab3b7b2b1b6b1ab1ab5Aa","unstructured":"[5] Nathan Jacobson. Basic Algebra I. 2nd edition. Dover Publications Inc., 2009."},{"key":"2021040808485936510_j_forma-2017-0018_ref_006_w2aab3b7b2b1b6b1ab1ab6Aa","unstructured":"[6] Heinz L\u00fcneburg. Gruppen, Ringe, K\u00f6rper: Die grundlegenden Strukturen der Algebra. Oldenbourg Verlag, 1990."},{"key":"2021040808485936510_j_forma-2017-0018_ref_007_w2aab3b7b2b1b6b1ab1ab7Aa","unstructured":"[7] Knut Radbruch. Algebra I. Lecture Notes, University of Kaiserslautern, Germany, 1991."},{"key":"2021040808485936510_j_forma-2017-0018_ref_008_w2aab3b7b2b1b6b1ab1ab8Aa","unstructured":"[8] Christoph Schwarzweller and Agnieszka Rowi\u0144ska-Schwarzweller. Schur\u2019s theorem on the stability of networks. Formalized Mathematics, 14(4):135\u2013142, 2006. doi:10.2478\/v10037-006-0017-9.10.2478\/v10037-006-0017-9"},{"key":"2021040808485936510_j_forma-2017-0018_ref_009_w2aab3b7b2b1b6b1ab1ab9Aa","unstructured":"[9] Christoph Schwarzweller, Artur Korni\u0142owicz, and Agnieszka Rowi\u0144ska-Schwarzweller. Some algebraic properties of polynomial rings. Formalized Mathematics, 24(3):227\u2013237, 2016. doi:10.1515\/forma-2016-0019.10.1515\/forma-2016-0019"}],"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/content.sciendo.com\/view\/journals\/forma\/25\/3\/article-p185.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.sciendo.com\/article\/10.1515\/forma-2017-0018","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,4,9]],"date-time":"2021-04-09T05:36:50Z","timestamp":1617946610000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.sciendo.com\/article\/10.1515\/forma-2017-0018"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,10,1]]},"references-count":9,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2017,12,19]]},"published-print":{"date-parts":[[2017,10,1]]}},"alternative-id":["10.1515\/forma-2017-0018"],"URL":"https:\/\/doi.org\/10.1515\/forma-2017-0018","relation":{},"ISSN":["1898-9934","1426-2630"],"issn-type":[{"value":"1898-9934","type":"electronic"},{"value":"1426-2630","type":"print"}],"subject":[],"published":{"date-parts":[[2017,10,1]]}}}