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http://hdl.handle.net/11320/14244
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Pole DC | Wartość | Język |
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dc.contributor.author | Schwarzweller, Christoph | - |
dc.contributor.author | Burgoa, Sara | - |
dc.date.accessioned | 2022-12-29T10:07:24Z | - |
dc.date.available | 2022-12-29T10:07:24Z | - |
dc.date.issued | 2022 | - |
dc.identifier.citation | Formalized Mathematics, Volume 30, Issue 1, Pages 23-30 | pl |
dc.identifier.issn | 1426-2630 | - |
dc.identifier.uri | http://hdl.handle.net/11320/14244 | - |
dc.description.abstract | In [11] the existence (and uniqueness) of splitting fields has been formalized. In this article we apply this result by providing splitting fields for the polynomials X² − 2, X³ − 1, X² + X + 1 and X³ − 2 over Q using the Mizar [2], [1] formalism. We also compute the degrees and bases for these splitting fields, which requires some additional registrations to adopt types properly. The main result, however, is that the polynomial X³ − 2 does not split over Q(∛2). Because X³ − 2 obviously has a root over Q(∛2), this shows that the field extension Q(∛2) is not normal over Q [3], [4], [5] and [7]. | pl |
dc.language.iso | en | pl |
dc.publisher | DeGruyter Open | pl |
dc.relation.uri | https://creativecommons.org/licenses/by-sa/3.0/ | pl |
dc.rights | Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0) | pl |
dc.rights.uri | https://creativecommons.org/licenses/by-sa/3.0/ | pl |
dc.subject | splitting fields | pl |
dc.subject | rational polynomials | pl |
dc.title | Splitting Fields for the Rational Polynomials X²−2, X²+X+1, X³−1, and X³−2 | pl |
dc.type | Article | pl |
dc.rights.holder | © 2022 The Author(s) | pl |
dc.rights.holder | CC BY-SA 3.0 license | pl |
dc.identifier.doi | 10.2478/forma-2022-0003 | - |
dc.description.Affiliation | Christoph Schwarzweller - Institute of Informatics, University of Gdańsk, Poland | pl |
dc.description.Affiliation | Sara Burgoa - Weston, Florida United States of America | pl |
dc.description.references | Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, 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 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8_17. | pl |
dc.description.references | Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6. | pl |
dc.description.references | Nathan Jacobson. Basic Algebra I. Dover Books on Mathematics, 1985. | pl |
dc.description.references | Serge Lang. Algebra. Springer Verlag, 2002 (Revised Third Edition). | pl |
dc.description.references | Heinz Lüneburg. Gruppen, Ringe, Körper: Die grundlegenden Strukturen der Algebra. Oldenbourg Verlag, 1999. | pl |
dc.description.references | Anna Justyna Milewska. The field of complex numbers. Formalized Mathematics, 9(2): 265–269, 2001. | pl |
dc.description.references | Knut Radbruch. Algebra I. Lecture Notes, University of Kaiserslautern, Germany, 1991. | pl |
dc.description.references | Christoph Schwarzweller. Field extensions and Kronecker’s construction. Formalized Mathematics, 27(3):229–235, 2019. doi:10.2478/forma-2019-0022. | pl |
dc.description.references | Christoph Schwarzweller. Renamings and a condition-free formalization of Kronecker’s construction. Formalized Mathematics, 28(2):129–135, 2020. doi:10.2478/forma-2020-0012. | pl |
dc.description.references | Christoph Schwarzweller. Ring and field adjunctions, algebraic elements and minimal polynomials. Formalized Mathematics, 28(3):251–261, 2020. doi:10.2478/forma-2020-0022. | pl |
dc.description.references | Christoph Schwarzweller. Splitting fields. Formalized Mathematics, 29(3):129–139, 2021. doi:10.2478/forma-2021-0013. | pl |
dc.description.references | Christoph Schwarzweller. On roots of polynomials and algebraically closed fields. Formalized Mathematics, 25(3):185–195, 2017. doi:10.1515/forma-2017-0018. | pl |
dc.description.references | Christoph Schwarzweller and Agnieszka Rowińska-Schwarzweller. Algebraic extensions. Formalized Mathematics, 29(1):39–48, 2021. doi:10.2478/forma-2021-0004. | pl |
dc.identifier.eissn | 1898-9934 | - |
dc.description.volume | 30 | pl |
dc.description.issue | 1 | pl |
dc.description.firstpage | 23 | pl |
dc.description.lastpage | 30 | pl |
dc.identifier.citation2 | Formalized Mathematics | pl |
dc.identifier.orcid | 0000-0001-9587-8737 | - |
Występuje w kolekcji(ach): | Formalized Mathematics, 2022, Volume 30, Issue 1 |
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