Algebraic Extensions

Pole DC	Wartość	Język
dc.contributor.author	Schwarzweller, Christoph	-
dc.contributor.author	Rowińska-Schwarzweller, Agnieszka	-
dc.date.accessioned	2021-08-30T08:49:21Z	-
dc.date.available	2021-08-30T08:49:21Z	-
dc.date.issued	2021	-
dc.identifier.citation	Formalized Mathematics, Volume 29, Issue 1, Pages 39-47	pl
dc.identifier.issn	1426-2630	-
dc.identifier.uri	http://hdl.handle.net/11320/11410	-
dc.description.abstract	In this article we further develop field theory in Mizar [1], [2], [3] towards splitting fields. We deal with algebraic extensions [4], [5]: a field extension E of a field F is algebraic, if every element of E is algebraic over F. We prove amongst others that finite extensions are algebraic and that field extensions generated by a finite set of algebraic elements are finite. From this immediately follows that field extensions generated by roots of a polynomial over F are both finite and algebraic. We also define the field of algebraic elements of E over F and show that this field is an intermediate field of E\|F.	pl
dc.language.iso	en	pl
dc.publisher	DeGruyter Open	pl
dc.rights	Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0)	-
dc.rights.uri	https://creativecommons.org/licenses/by-sa/3.0/	-
dc.subject	algebraic extensions	pl
dc.subject	finite extensions	pl
dc.subject	field of algebraic numbers	pl
dc.title	Algebraic 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-2021-0004	-
dc.description.Affiliation	Christoph Schwarzweller - Institute of Informatics, University of Gdansk, Poland	pl
dc.description.references	Grzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pak, 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 Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pak. 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
dc.description.references	Serge Lang. Algebra. Springer, 3rd edition, 2005.	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.identifier.eissn	1898-9934	-
dc.description.volume	29	pl
dc.description.issue	1	pl
dc.description.firstpage	39	pl
dc.description.lastpage	47	pl
dc.identifier.citation2	Formalized Mathematics	pl
dc.identifier.orcid	0000-0001-9587-8737	-
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