{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T15:53:19Z","timestamp":1777477999217,"version":"3.51.4"},"reference-count":14,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2023,9,1]],"date-time":"2023-09-01T00:00:00Z","timestamp":1693526400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-sa\/3.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,9,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>\n                    In this article we continue the formalization of field theory in Mizar [1], [2], [4], [3]. We introduce normal extensions: an (algebraic) extension\n                    <jats:italic>E<\/jats:italic>\n                    of\n                    <jats:italic>F<\/jats:italic>\n                    is normal if every polynomial of\n                    <jats:italic>F<\/jats:italic>\n                    that has a root in\n                    <jats:italic>E<\/jats:italic>\n                    already splits in\n                    <jats:italic>E<\/jats:italic>\n                    . We proved characterizations (for finite extensions) by minimal polynomials [7], splitting fields, and fixing monomorphisms [6], [5]. This required extending results from [11] and [12], in particular that\n                    <jats:italic>F<\/jats:italic>\n                    [\n                    <jats:italic>T<\/jats:italic>\n                    ] = {\n                    <jats:italic>p<\/jats:italic>\n                    (\n                    <jats:italic>a<\/jats:italic>\n                    <jats:sub>1<\/jats:sub>\n                    , . . .\n                    <jats:italic>\n                      a\n                      <jats:sub>n<\/jats:sub>\n                    <\/jats:italic>\n                    ) |\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>\n                      a\n                      <jats:sub>i<\/jats:sub>\n                    <\/jats:italic>\n                    \u2208\n                    <jats:italic>T<\/jats:italic>\n                    } and\n                    <jats:italic>F<\/jats:italic>\n                    (\n                    <jats:italic>T<\/jats:italic>\n                    ) =\n                    <jats:italic>F<\/jats:italic>\n                    [\n                    <jats:italic>T<\/jats:italic>\n                    ] for finite algebraic\n                    <jats:italic>T<\/jats:italic>\n                    \u2286\n                    <jats:italic>E<\/jats:italic>\n                    . We also provided the counterexample that \ud835\udcac(\u221b2) is not normal over \ud835\udcac (compare [13]).\n                  <\/jats:p>","DOI":"10.2478\/forma-2023-0011","type":"journal-article","created":{"date-parts":[[2023,11,1]],"date-time":"2023-11-01T13:24:13Z","timestamp":1698845053000},"page":"121-130","source":"Crossref","is-referenced-by-count":5,"title":["Normal Extensions"],"prefix":"10.2478","volume":"31","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":[[2023,11,1]]},"reference":[{"key":"2026042801414635798_j_forma-2023-0011_ref_001","doi-asserted-by":"crossref","unstructured":"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.","DOI":"10.1007\/978-3-319-20615-8_17"},{"key":"2026042801414635798_j_forma-2023-0011_ref_002","doi-asserted-by":"crossref","unstructured":"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.","DOI":"10.1007\/s10817-017-9440-6"},{"key":"2026042801414635798_j_forma-2023-0011_ref_003","doi-asserted-by":"crossref","unstructured":"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.","DOI":"10.1007\/s10817-015-9345-1"},{"key":"2026042801414635798_j_forma-2023-0011_ref_004","doi-asserted-by":"crossref","unstructured":"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":"2026042801414635798_j_forma-2023-0011_ref_005","unstructured":"Serge Lang. Algebra. Springer Verlag, 2002 (Revised Third Edition)."},{"key":"2026042801414635798_j_forma-2023-0011_ref_006","unstructured":"Knut Radbruch. Algebra I. Lecture Notes, University of Kaiserslautern, Germany, 1991."},{"key":"2026042801414635798_j_forma-2023-0011_ref_007","doi-asserted-by":"crossref","unstructured":"Piotr Rudnicki, Christoph Schwarzweller, and Andrzej Trybulec. Commutative algebra in the Mizar system. Journal of Symbolic Computation, 32(1\/2):143\u2013169, 2001. doi:10.1006\/jsco.2001.0456.","DOI":"10.1006\/jsco.2001.0456"},{"key":"2026042801414635798_j_forma-2023-0011_ref_008","doi-asserted-by":"crossref","unstructured":"Christoph Schwarzweller. Artin\u2019s theorem towards the existence of algebraic closures. Formalized Mathematics, 30(3):199\u2013207, 2022. doi:10.2478\/forma-2022-0014.","DOI":"10.2478\/forma-2022-0014"},{"key":"2026042801414635798_j_forma-2023-0011_ref_009","doi-asserted-by":"crossref","unstructured":"Christoph Schwarzweller. Existence and uniqueness of algebraic closures. Formalized Mathematics, 30(4):281\u2013294, 2022. doi:10.2478\/forma-2022-0022.","DOI":"10.2478\/forma-2022-0022"},{"key":"2026042801414635798_j_forma-2023-0011_ref_010","doi-asserted-by":"crossref","unstructured":"Christoph Schwarzweller. Field extensions and Kronecker\u2019s construction. Formalized Mathematics, 27(3):229\u2013235, 2019. doi:10.2478\/forma-2019-0022.","DOI":"10.2478\/forma-2019-0022"},{"key":"2026042801414635798_j_forma-2023-0011_ref_011","doi-asserted-by":"crossref","unstructured":"Christoph Schwarzweller. Ring and field adjunctions, algebraic elements and minimal polynomials. Formalized Mathematics, 28(3):251\u2013261, 2020. doi:10.2478\/forma-2020-0022.","DOI":"10.2478\/forma-2020-0022"},{"key":"2026042801414635798_j_forma-2023-0011_ref_012","doi-asserted-by":"crossref","unstructured":"Christoph Schwarzweller. Splitting fields. 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Formalized Mathematics, 29(1):39\u201348, 2021. doi:10.2478\/forma-2021-0004.","DOI":"10.2478\/forma-2021-0004"}],"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/reference-global.com\/pdf\/10.2478\/forma-2023-0011","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T14:00:58Z","timestamp":1777384858000},"score":1,"resource":{"primary":{"URL":"https:\/\/reference-global.com\/article\/10.2478\/forma-2023-0011"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,1]]},"references-count":14,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,11,1]]},"published-print":{"date-parts":[[2023,9,1]]}},"alternative-id":["10.2478\/forma-2023-0011"],"URL":"https:\/\/doi.org\/10.2478\/forma-2023-0011","relation":{},"ISSN":["1898-9934"],"issn-type":[{"value":"1898-9934","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,9,1]]}}}