{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T15:51:25Z","timestamp":1777477885271,"version":"3.51.4"},"reference-count":12,"publisher":"Walter de Gruyter GmbH","issue":"4","license":[{"start":{"date-parts":[[2021,12,1]],"date-time":"2021-12-01T00:00:00Z","timestamp":1638316800000},"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":[[2021,12,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>\n                    In this article we further develop field theory [6], [7], [12] in Mizar [1], [2], [3]: we deal with quadratic polynomials and quadratic extensions [5], [4]. First we introduce quadratic polynomials, their discriminants and prove the midnight formula. Then we show that - in case the discriminant of\n                    <jats:italic>p<\/jats:italic>\n                    being non square - adjoining a root of\n                    <jats:italic>p<\/jats:italic>\n                    \u2019s discriminant results in a splitting field of\n                    <jats:italic>p<\/jats:italic>\n                    . Finally we prove that these are the only field extensions of degree 2, e.g. that an extension\n                    <jats:italic>E<\/jats:italic>\n                    of\n                    <jats:italic>F<\/jats:italic>\n                    is quadratic if and only if there is a non square Element\n                    <jats:italic>a<\/jats:italic>\n                    \u2208\n                    <jats:italic>F<\/jats:italic>\n                    such that\n                    <jats:italic>E<\/jats:italic>\n                    and  (\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_forma-2021-0021_eq_001.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\">\n                          <m:mrow>\n                            <m:mi>F<\/m:mi>\n                            <m:msqrt>\n                              <m:mi>a<\/m:mi>\n                            <\/m:msqrt>\n                          <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>F\\sqrt a<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    ) are isomorphic over\n                    <jats:italic>F<\/jats:italic>\n                    .\n                  <\/jats:p>","DOI":"10.2478\/forma-2021-0021","type":"journal-article","created":{"date-parts":[[2022,7,9]],"date-time":"2022-07-09T16:34:22Z","timestamp":1657384462000},"page":"229-240","source":"Crossref","is-referenced-by-count":1,"title":["Quadratic Extensions"],"prefix":"10.2478","volume":"29","author":[{"given":"Christoph","family":"Schwarzweller","sequence":"first","affiliation":[{"name":"Institute of Informatics , University of Gda\u0144sk , Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Agnieszka","family":"Rowi\u0144ska-Schwarzweller","sequence":"additional","affiliation":[{"name":"Sopot , Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2022,7,9]]},"reference":[{"key":"2026042801423308690_j_forma-2021-0021_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.","DOI":"10.1007\/978-3-319-20615-8_17"},{"key":"2026042801423308690_j_forma-2021-0021_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\/s10817-017-9440-6.604425130069070","DOI":"10.1007\/s10817-017-9440-6"},{"key":"2026042801423308690_j_forma-2021-0021_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.","DOI":"10.15439\/2016F520"},{"key":"2026042801423308690_j_forma-2021-0021_ref_004","unstructured":"[4] Nathan Jacobson. Basic Algebra I. Dover Books on Mathematics, 1985."},{"key":"2026042801423308690_j_forma-2021-0021_ref_005","unstructured":"[5] Serge Lang. Algebra. Springer Verlag, 2002 (Revised Third Edition)."},{"key":"2026042801423308690_j_forma-2021-0021_ref_006","doi-asserted-by":"crossref","unstructured":"[6] Heinz L\u00fcneburg. Gruppen, Ringe, K\u00f6rper: Die grundlegenden Strukturen der Algebra. Oldenbourg Verlag, 1999.10.1524\/9783486599022","DOI":"10.1524\/9783486599022"},{"key":"2026042801423308690_j_forma-2021-0021_ref_007","unstructured":"[7] Knut Radbruch. Algebra I. Lecture Notes, University of Kaiserslautern, Germany, 1991."},{"key":"2026042801423308690_j_forma-2021-0021_ref_008","doi-asserted-by":"crossref","unstructured":"[8] 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":"2026042801423308690_j_forma-2021-0021_ref_009","doi-asserted-by":"crossref","unstructured":"[9] Christoph Schwarzweller. Formally real fields. Formalized Mathematics, 25(4):249\u2013259, 2017. doi:10.1515\/forma-2017-0024.","DOI":"10.1515\/forma-2017-0024"},{"key":"2026042801423308690_j_forma-2021-0021_ref_010","doi-asserted-by":"crossref","unstructured":"[10] Christoph Schwarzweller. On roots of polynomials and algebraically closed fields. Formalized Mathematics, 25(3):185\u2013195, 2017. doi:10.1515\/forma-2017-0018.","DOI":"10.1515\/forma-2017-0018"},{"key":"2026042801423308690_j_forma-2021-0021_ref_011","doi-asserted-by":"crossref","unstructured":"[11] Christoph Schwarzweller and Artur Korni\u0142owicz. Characteristic of rings. Prime fields. Formalized Mathematics, 23(4):333\u2013349, 2015. doi:10.1515\/forma-2015-0027.","DOI":"10.1515\/forma-2015-0027"},{"key":"2026042801423308690_j_forma-2021-0021_ref_012","doi-asserted-by":"crossref","unstructured":"[12] Steven H. Weintraub. Galois Theory. Springer-Verlag, 2 edition, 2009.10.1007\/978-0-387-87575-0","DOI":"10.1007\/978-0-387-87575-0"}],"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/reference-global.com\/pdf\/10.2478\/forma-2021-0021","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T14:00:14Z","timestamp":1777384814000},"score":1,"resource":{"primary":{"URL":"https:\/\/reference-global.com\/article\/10.2478\/forma-2021-0021"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,1]]},"references-count":12,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2022,7,9]]},"published-print":{"date-parts":[[2021,12,1]]}},"alternative-id":["10.2478\/forma-2021-0021"],"URL":"https:\/\/doi.org\/10.2478\/forma-2021-0021","relation":{},"ISSN":["1898-9934"],"issn-type":[{"value":"1898-9934","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,12,1]]}}}