{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T12:38:02Z","timestamp":1777466282875,"version":"3.51.4"},"reference-count":13,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2024,8,1]],"date-time":"2024-08-01T00:00:00Z","timestamp":1722470400000},"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":[[2024,8,1]]},"abstract":"Summary<\/jats:title>\n \n We continue the formalization of field theory in Mizar [2], [3], [4]. We introduce separability of polynomials and field extensions: a polynomial is separable, if it has no multiple roots in its splitting field; an algebraic extension\n E<\/jats:italic>\n of\n F<\/jats:italic>\n is separable, if the minimal polynomial of each\n a<\/jats:italic>\n \u2208\n E<\/jats:italic>\n is separable. We prove among others that a polynomial\n q<\/jats:italic>\n (\n X<\/jats:italic>\n ) is separable if and only if the gcd of\n q<\/jats:italic>\n (\n X<\/jats:italic>\n ) and its (formal) derivation equals 1 \u2013 and that a irreducible polynomial\n q<\/jats:italic>\n (\n X<\/jats:italic>\n ) is separable if and only if its derivation is not 0 \u2013 and that\n q<\/jats:italic>\n (\n X<\/jats:italic>\n ) is separable if and only if the number of\n q<\/jats:italic>\n (\n X<\/jats:italic>\n )\u2019s roots in some field extension equals the degree of\n q<\/jats:italic>\n (\n X<\/jats:italic>\n ).\n <\/jats:p>\n \n A field\n F<\/jats:italic>\n is called perfect if all irreducible polynomials over\n F<\/jats:italic>\n are separable, and as a consequence every algebraic extension of\n F<\/jats:italic>\n is separable. Every field with characteristic 0 is perfect [13]. To also consider separability in fields with prime characteristic\n p<\/jats:italic>\n we define the rings\n \n R\n p<\/jats:sup>\n <\/jats:italic>\n =\n \n { a\n p<\/jats:sup>\n <\/jats:italic>\n | a<\/jats:italic>\n \u2208\n R}<\/jats:italic>\n and the polynomials\n \n X\n n<\/jats:sup>\n <\/jats:italic>\n \u2212 a<\/jats:italic>\n for\n a<\/jats:italic>\n \u2208\n R<\/jats:italic>\n . Then we show that a field\n F<\/jats:italic>\n with prime characteristic\n p<\/jats:italic>\n is separable if and only if\n F<\/jats:italic>\n =\n \n F\n p<\/jats:sup>\n <\/jats:italic>\n and that finite fields are perfect. Finally we prove that for fields\n F<\/jats:italic>\n \u2286\n K<\/jats:italic>\n \u2286\n E<\/jats:italic>\n where\n E<\/jats:italic>\n is a separable extension of\n F<\/jats:italic>\n both\n E<\/jats:italic>\n is separable over\n K<\/jats:italic>\n and\n K<\/jats:italic>\n is separable over\n F<\/jats:italic>\n .\n <\/jats:p>","DOI":"10.2478\/forma-2024-0003","type":"journal-article","created":{"date-parts":[[2024,9,2]],"date-time":"2024-09-02T06:28:54Z","timestamp":1725258534000},"page":"33-46","source":"Crossref","is-referenced-by-count":0,"title":["Separable Polynomials and Separable Extensions"],"prefix":"10.2478","volume":"32","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":[[2024,8,31]]},"reference":[{"key":"2026042801390485079_j_forma-2024-0003_ref_001","unstructured":"Broderick Arneson and Piotr Rudnicki. Primitive roots of unity and cyclotomic polynomials. Formalized Mathematics, 12(1):59\u201367, 2004."},{"key":"2026042801390485079_j_forma-2024-0003_ref_002","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":"2026042801390485079_j_forma-2024-0003_ref_003","doi-asserted-by":"crossref","unstructured":"Grzegorz Bancerek, Czes\u0142aw Byli\u0144ski, Adam Grabowski, Artur Korni\u0142owicz, Roman Matuszewski, Adam Naumowicz, and Karol P\u0105k. 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Lecture Notes, University of Kaiserslautern, Germany, 1991."},{"key":"2026042801390485079_j_forma-2024-0003_ref_011","unstructured":"Christoph Schwarzweller. The binomial theorem for algebraic structures. Formalized Mathematics, 9(3):559\u2013564, 2001."},{"key":"2026042801390485079_j_forma-2024-0003_ref_012","doi-asserted-by":"crossref","unstructured":"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":"2026042801390485079_j_forma-2024-0003_ref_013","doi-asserted-by":"crossref","unstructured":"Christoph Schwarzweller and Agnieszka Rowi\u0144ska-Schwarzweller. Simple extensions. 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