Quadratic Extensions

Pole DC	Wartość	Język
dc.contributor.author	Schwarzweller, Christoph	-
dc.contributor.author	Rowińska-Schwarzweller, Agnieszka	-
dc.date.accessioned	2022-07-25T06:47:08Z	-
dc.date.available	2022-07-25T06:47:08Z	-
dc.date.issued	2021	-
dc.identifier.citation	Formalized Mathematics, Volume 29, Issue 4, Pages 229-240	pl
dc.identifier.issn	1426-2630	-
dc.identifier.uri	http://hdl.handle.net/11320/13667	-
dc.description.abstract	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 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 F.	pl
dc.language.iso	en	pl
dc.publisher	DeGruyter Open	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	field extensions	pl
dc.subject	quadratic polynomials	pl
dc.subject	quadratic extensions	pl
dc.title	Quadratic Extensions	pl
dc.type	Article	pl
dc.rights.holder	© 2021 University of Białymstoku	pl
dc.rights.holder	CC-BY-SA License ver. 3.0 or later	pl
dc.identifier.doi	10.2478/forma-2021-0021	-
dc.description.Affiliation	Christoph Schwarzweller - Institute of Informatics, University of Gdańsk, Poland	pl
dc.description.Affiliation	Agnieszka Rowińska-Schwarzweller - Sopot, Poland	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	Adam 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:10.15439/2016F520.	pl
dc.description.references	Nathan Jacobson. Basic Algebra I. Dover Books on Mathematics, 1985.	pl
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dc.description.references	Knut Radbruch. Algebra I. Lecture Notes, University of Kaiserslautern, Germany, 1991.	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. Formally real fields. Formalized Mathematics, 25(4):249–259, 2017. doi:10.1515/forma-2017-0024.	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 Artur Korniłowicz. Characteristic of rings. Prime fields. Formalized Mathematics, 23(4):333–349, 2015. doi:10.1515/forma-2015-0027.	pl
dc.description.references	Steven H. Weintraub. Galois Theory. Springer-Verlag, 2 edition, 2009.	pl
dc.identifier.eissn	1898-9934	-
dc.description.volume	29	pl
dc.description.issue	4	pl
dc.description.firstpage	229	pl
dc.description.lastpage	240	pl
dc.identifier.citation2	Formalized Mathematics	pl
dc.identifier.orcid	0000-0001-9587-8737	-
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