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dc.contributor.authorSchwarzweller, Christoph-
dc.contributor.authorRowińska-Schwarzweller, Agnieszka-
dc.identifier.citationFormalized Mathematics, Volume 29, Issue 4, Pages 229-240pl
dc.description.abstractIn 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 p being non square - adjoining a root of p’s discriminant results in a splitting field of p. Finally we prove that these are the only field extensions of degree 2, e.g. that an extension E of F is quadratic if and only if there is a non square Element a ∈ F such that E and F(√a) are isomorphic over
dc.publisherDeGruyter Openpl
dc.rightsAttribution-ShareAlike 3.0 Unported (CC BY-SA 3.0)pl
dc.subjectfield extensionspl
dc.subjectquadratic polynomialspl
dc.subjectquadratic extensionspl
dc.titleQuadratic Extensionspl
dc.rights.holder© 2021 University of Białymstokupl
dc.rights.holderCC-BY-SA License ver. 3.0 or laterpl
dc.description.AffiliationChristoph Schwarzweller - Institute of Informatics, University of Gdańsk, Polandpl
dc.description.AffiliationAgnieszka Rowińska-Schwarzweller - Sopot, Polandpl
dc.description.referencesGrzegorz 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
dc.description.referencesGrzegorz 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/
dc.description.referencesAdam Grabowski, Artur Korniłowicz, 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 363–371, 2016. doi:10.15439/
dc.description.referencesNathan Jacobson. Basic Algebra I. Dover Books on Mathematics,
dc.description.referencesSerge Lang. Algebra. Springer Verlag, 2002 (Revised Third Edition).pl
dc.description.referencesHeinz Luneburg. Gruppen, Ringe, K¨orper: Die grundlegenden Strukturen der Algebra. Oldenbourg Verlag,
dc.description.referencesKnut Radbruch. Algebra I. Lecture Notes, University of Kaiserslautern, Germany,
dc.description.referencesChristoph Schwarzweller. Ring and field adjunctions, algebraic elements and minimal polynomials. Formalized Mathematics, 28(3):251–261, 2020. doi:10.2478/
dc.description.referencesChristoph Schwarzweller. Formally real fields. Formalized Mathematics, 25(4):249–259, 2017. doi:10.1515/
dc.description.referencesChristoph Schwarzweller. On roots of polynomials and algebraically closed fields. Formalized Mathematics, 25(3):185–195, 2017. doi:10.1515/
dc.description.referencesChristoph Schwarzweller and Artur Korniłowicz. Characteristic of rings. Prime fields. Formalized Mathematics, 23(4):333–349, 2015. doi:10.1515/
dc.description.referencesSteven H. Weintraub. Galois Theory. Springer-Verlag, 2 edition,
dc.identifier.citation2Formalized Mathematicspl
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