{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,15]],"date-time":"2026-06-15T01:08:27Z","timestamp":1781485707616,"version":"3.54.1"},"reference-count":6,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2021,4,1]],"date-time":"2021-04-01T00:00:00Z","timestamp":1617235200000},"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":[[2021,4,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>\n                    In this article we further develop field theory in Mizar [1], [2], [3] towards splitting fields. We deal with algebraic extensions [4], [5]: a field extension\n                    <jats:italic>E<\/jats:italic>\n                    of a field\n                    <jats:italic>F<\/jats:italic>\n                    is algebraic, if every element of\n                    <jats:italic>E<\/jats:italic>\n                    is algebraic over\n                    <jats:italic>F<\/jats:italic>\n                    . We prove amongst others that finite extensions are algebraic and that field extensions generated by a finite set of algebraic elements are finite. From this immediately follows that field extensions generated by roots of a polynomial over\n                    <jats:italic>F<\/jats:italic>\n                    are both finite and algebraic. We also define the field of algebraic elements of\n                    <jats:italic>E<\/jats:italic>\n                    over\n                    <jats:italic>F<\/jats:italic>\n                    and show that this field is an intermediate field of\n                    <jats:italic>E|F.<\/jats:italic>\n                  <\/jats:p>","DOI":"10.2478\/forma-2021-0004","type":"journal-article","created":{"date-parts":[[2021,8,27]],"date-time":"2021-08-27T11:48:43Z","timestamp":1630064923000},"page":"39-47","source":"Crossref","is-referenced-by-count":5,"title":["Algebraic Extensions"],"prefix":"10.2478","volume":"29","author":[{"given":"Christoph","family":"Schwarzweller","sequence":"first","affiliation":[{"name":"Institute of Informatics , University of Gda\u0144sk , Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Agnieszka","family":"Rowi\u0144ska-Schwarzweller","sequence":"additional","affiliation":[{"name":"Sopot , Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"374","published-online":{"date-parts":[[2021,8,26]]},"reference":[{"key":"2026042801401493549_j_forma-2021-0004_ref_001","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":"2026042801401493549_j_forma-2021-0004_ref_002","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":"2026042801401493549_j_forma-2021-0004_ref_003","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\/2016F520.10.15439\/2016F520","DOI":"10.15439\/2016F520"},{"key":"2026042801401493549_j_forma-2021-0004_ref_004","unstructured":"[4] Nathan Jacobson. Basic Algebra I. Dover Books on Mathematics, 1985."},{"key":"2026042801401493549_j_forma-2021-0004_ref_005","unstructured":"[5] Serge Lang. Algebra. Springer, 3rd edition, 2005."},{"key":"2026042801401493549_j_forma-2021-0004_ref_006","doi-asserted-by":"crossref","unstructured":"[6] Christoph Schwarzweller. Ring and field adjunctions, algebraic elements and minimal polynomials. Formalized Mathematics, 28(3):251\u2013261, 2020. doi:10.2478\/forma-2020-0022.10.2478\/forma-2020-0022","DOI":"10.2478\/forma-2020-0022"}],"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/reference-global.com\/pdf\/10.2478\/forma-2021-0004","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T13:39:27Z","timestamp":1777383567000},"score":1,"resource":{"primary":{"URL":"https:\/\/reference-global.com\/article\/10.2478\/forma-2021-0004"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4,1]]},"references-count":6,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,8,26]]},"published-print":{"date-parts":[[2021,4,1]]}},"alternative-id":["10.2478\/forma-2021-0004"],"URL":"https:\/\/doi.org\/10.2478\/forma-2021-0004","relation":{},"ISSN":["1898-9934"],"issn-type":[{"value":"1898-9934","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,4,1]]}}}