Simple Extensions

Pole DC	Wartość	Język
dc.contributor.author	Schwarzweller, Christoph	-
dc.contributor.author	Rowińska-Schwarzweller, Agnieszka	-
dc.date.accessioned	2024-01-26T09:50:48Z	-
dc.date.available	2024-01-26T09:50:48Z	-
dc.date.issued	2023	-
dc.identifier.citation	Formalized Mathematics, Volume 31, Issue 1, Pages 287-298	pl
dc.identifier.issn	1426-2630	-
dc.identifier.uri	http://hdl.handle.net/11320/15862	-
dc.description.abstract	In this article we continue the formalization of field theory in Mizar. We introduce simple extensions: an extension E of F is simple if E is generated over F by a single element of E, that is E = F(a) for some a ∈ E. First, we prove that a finite extension E of F is simple if and only if there are only finitely many intermediate fields between E and F [7]. Second, we show that finite extensions of a field F with characteristic 0 are always simple [1]. For this we had to prove, that irreducible polynomials over F have single roots only, which required extending results on divisibility and gcds of polynomials [14], [13] and formal derivation of polynomials [15].	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 theory	pl
dc.subject	intermediate field	pl
dc.subject	simple extension	pl
dc.subject	primitive element	pl
dc.title	Simple Extensions	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-2023-0023	-
dc.description.Affiliation	Christoph Schwarzweller - Institute of Informatics, University of Gdańsk, Poland	pl
dc.description.Affiliation	Agnieszka Rowińska-Schwarzweller - Institute of Informatics, University of Gdańsk, Poland	pl
dc.description.references	Andreas Gathmann. Einf¨uhrung in die Algebra. Lecture Notes, University of Kaiserslautern, Germany, 2011.	pl
dc.description.references	Adam Grabowski and Christoph Schwarzweller. Translating mathematical vernacular into knowledge repositories. In Michael Kohlhase, editor, Mathematical Knowledge Management, volume 3863 of Lecture Notes in Computer Science, pages 49–64. Springer, 2006. doi:10.1007/11618027 4. 4th International Conference on Mathematical Knowledge Management, Bremen, Germany, MKM 2005, July 15–17, 2005, Revised Selected Papers.	pl
dc.description.references	Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Mizar in a nutshell. Journal of Formalized Reasoning, 3(2):153–245, 2010.	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	Artur Korniłowicz. Flexary connectives in Mizar. Computer Languages, Systems & Structures, 44:238–250, December 2015. doi:10.1016/j.cl.2015.07.002.	pl
dc.description.references	Serge Lang. Algebra. PWN, Warszawa, 1984.	pl
dc.description.references	Serge Lang. Algebra. Springer Verlag, 2002 (Revised Third Edition).	pl
dc.description.references	Heinz L¨uneburg. Gruppen, Ringe, K¨orper: Die grundlegenden Strukturen der Algebra. Oldenbourg Verlag, 1999.	pl
dc.description.references	Christoph Schwarzweller. Normal extensions. Formalized Mathematics, 31(1):121–130, 2023. doi:10.2478/forma-2023-0011.	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 poynomials. 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, Artur Korniłowicz, and Agnieszka Rowińska-Schwarzweller. Some algebraic properties of polynomial rings. Formalized Mathematics, 24(3):227–237, 2016. doi:10.1515/forma-2016-0019.	pl
dc.description.references	Yasushige Watase. Derivation of commutative rings and the Leibniz formula for power of derivation. Formalized Mathematics, 29(1):1–8, 2021. doi:10.2478/forma-2021-0001.	pl
dc.identifier.eissn	1898-9934	-
dc.description.volume	31	pl
dc.description.issue	1	pl
dc.description.firstpage	287	pl
dc.description.lastpage	298	pl
dc.identifier.citation2	Formalized Mathematics	pl
dc.identifier.orcid	0000-0001-9587-8737	-
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