{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T14:44:39Z","timestamp":1777473879297,"version":"3.51.4"},"reference-count":8,"publisher":"Walter de Gruyter GmbH","issue":"3","license":[{"start":{"date-parts":[[2020,10,1]],"date-time":"2020-10-01T00:00:00Z","timestamp":1601510400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,10,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>\n                    In [6], [7] we presented a formalization of Kronecker\u2019s construction of a field extension of a field\n                    <jats:italic>F<\/jats:italic>\n                    in which a given polynomial\n                    <jats:italic>p<\/jats:italic>\n                    \u2208\n                    <jats:italic>F<\/jats:italic>\n                    [\n                    <jats:italic>X<\/jats:italic>\n                    ]\n                    <jats:italic>\\F<\/jats:italic>\n                    has a root [4], [5], [3]. As a consequence for every field\n                    <jats:italic>F<\/jats:italic>\n                    and every polynomial there exists a field extension\n                    <jats:italic>E<\/jats:italic>\n                    of\n                    <jats:italic>F<\/jats:italic>\n                    in which\n                    <jats:italic>p<\/jats:italic>\n                    splits into linear factors. It is well-known that one gets the smallest such field extension \u2013 the splitting field of\n                    <jats:italic>p<\/jats:italic>\n                    \u2013 by adjoining the roots of\n                    <jats:italic>p<\/jats:italic>\n                    to\n                    <jats:italic>F<\/jats:italic>\n                    .\n                  <\/jats:p>\n                  <jats:p>\n                    In this article we start the Mizar formalization [1], [2] towards splitting fields: we define ring and field adjunctions, algebraic elements and minimal polynomials and prove a number of facts necessary to develop the theory of splitting fields, in particular that for an algebraic element\n                    <jats:italic>a<\/jats:italic>\n                    over\n                    <jats:italic>F<\/jats:italic>\n                    a basis of the vector space\n                    <jats:italic>F<\/jats:italic>\n                    (\n                    <jats:italic>a<\/jats:italic>\n                    ) over\n                    <jats:italic>F<\/jats:italic>\n                    is given by\n                    <jats:italic>a<\/jats:italic>\n                    <jats:sup>0<\/jats:sup>\n                    <jats:italic>\n                      , . . ., a\n                      <jats:sup>n\u2212<\/jats:sup>\n                    <\/jats:italic>\n                    <jats:sup>1<\/jats:sup>\n                    , where\n                    <jats:italic>n<\/jats:italic>\n                    is the degree of the minimal polynomial of\n                    <jats:italic>a<\/jats:italic>\n                    over\n                    <jats:italic>F<\/jats:italic>\n                    .\n                  <\/jats:p>","DOI":"10.2478\/forma-2020-0022","type":"journal-article","created":{"date-parts":[[2021,4,29]],"date-time":"2021-04-29T18:57:28Z","timestamp":1619722648000},"page":"251-261","source":"Crossref","is-referenced-by-count":7,"title":["Ring and Field Adjunctions, Algebraic Elements and Minimal Polynomials"],"prefix":"10.2478","volume":"28","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":[[2021,4,6]]},"reference":[{"key":"2026042801401056967_j_forma-2020-0022_ref_001_w2aab3b7b5b1b6b1ab1ab1Aa","doi-asserted-by":"crossref","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","DOI":"10.1007\/978-3-319-20615-8_17"},{"key":"2026042801401056967_j_forma-2020-0022_ref_002_w2aab3b7b5b1b6b1ab1ab2Aa","doi-asserted-by":"crossref","unstructured":"[2] Grzegorz Bancerek, Czes\u0142aw Byli\u0144ski, Adam Grabowski, Artur Korni\u0142owicz, Roman Matuszewski, Adam Naumowicz, and Karol P\u0105k. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9\u201332, 2018. doi:10.1007\/s10817-017-9440-6.10.1007\/s10817-017-9440-6604425130069070","DOI":"10.1007\/s10817-017-9440-6"},{"key":"2026042801401056967_j_forma-2020-0022_ref_003_w2aab3b7b5b1b6b1ab1ab3Aa","unstructured":"[3] Nathan Jacobson. Basic Algebra I. Dover Books on Mathematics, 1985."},{"key":"2026042801401056967_j_forma-2020-0022_ref_004_w2aab3b7b5b1b6b1ab1ab4Aa","doi-asserted-by":"crossref","unstructured":"[4] Heinz L\u00fcneburg. Gruppen, Ringe, K\u00f6rper: Die grundlegenden Strukturen der Algebra. Oldenbourg Verlag, 1999.10.1524\/9783486599022","DOI":"10.1524\/9783486599022"},{"key":"2026042801401056967_j_forma-2020-0022_ref_005_w2aab3b7b5b1b6b1ab1ab5Aa","unstructured":"[5] Knut Radbruch. Algebra I. Lecture Notes, University of Kaiserslautern, Germany, 1991."},{"key":"2026042801401056967_j_forma-2020-0022_ref_006_w2aab3b7b5b1b6b1ab1ab6Aa","doi-asserted-by":"crossref","unstructured":"[6] Christoph Schwarzweller. Renamings and a condition-free formalization of Kronecker\u2019s construction. Formalized Mathematics, 28(2):129\u2013135, 2020. doi:10.2478\/forma-2020-0012.10.2478\/forma-2020-0012","DOI":"10.2478\/forma-2020-0012"},{"key":"2026042801401056967_j_forma-2020-0022_ref_007_w2aab3b7b5b1b6b1ab1ab7Aa","doi-asserted-by":"crossref","unstructured":"[7] Christoph Schwarzweller. Representation matters: An unexpected property of polynomial rings and its consequences for formalizing abstract field theory. In M. Ganzha, L. Maciaszek, and M. Paprzycki, editors, Proceedings of the 2018 Federated Conference on Computer Science and Information Systems, volume 15 of Annals of Computer Science and Information Systems, pages 67\u201372. IEEE, 2018. doi:10.15439\/2018F88.10.15439\/2018F88","DOI":"10.15439\/2018F88"},{"key":"2026042801401056967_j_forma-2020-0022_ref_008_w2aab3b7b5b1b6b1ab1ab8Aa","doi-asserted-by":"crossref","unstructured":"[8] Yasushige Watase. Algebraic numbers. Formalized Mathematics, 24(4):291\u2013299, 2016. doi:10.1515\/forma-2016-0025.10.1515\/forma-2016-0025","DOI":"10.1515\/forma-2016-0025"}],"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/reference-global.com\/pdf\/10.2478\/forma-2020-0022","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T13:47:55Z","timestamp":1777384075000},"score":1,"resource":{"primary":{"URL":"https:\/\/reference-global.com\/article\/10.2478\/forma-2020-0022"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,10,1]]},"references-count":8,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2021,4,6]]},"published-print":{"date-parts":[[2020,10,1]]}},"alternative-id":["10.2478\/forma-2020-0022"],"URL":"https:\/\/doi.org\/10.2478\/forma-2020-0022","relation":{},"ISSN":["1898-9934","1426-2630"],"issn-type":[{"value":"1898-9934","type":"electronic"},{"value":"1426-2630","type":"print"}],"subject":[],"published":{"date-parts":[[2020,10,1]]}}}