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http://hdl.handle.net/11320/15867
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Pole DC | Wartość | Język |
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dc.contributor.author | Schwarzweller, Christoph | - |
dc.date.accessioned | 2024-01-26T11:54:48Z | - |
dc.date.available | 2024-01-26T11:54:48Z | - |
dc.date.issued | 2023 | - |
dc.identifier.citation | Formalized Mathematics, Volume 31, Issue 1, Pages 341-352 | pl |
dc.identifier.issn | 1426-2630 | - |
dc.identifier.uri | http://hdl.handle.net/11320/15867 | - |
dc.description.abstract | In this article we extend the algebraic theory of ordered fields [6], [8] in Mizar. We introduce extensions of orderings: if E is a field extension of F, then an ordering P of F extends to E, if there exists an ordering O of E containing P. We first prove some necessary and sufficient conditions for P being extendable to E, in particular that P extends to E if and only if the set QS E := {∑a ∗ b²| a ∈ P, b ∈ E} is a preordering of E – or equivalently if and only if −1 ∈/ QS E. Then we show for non-square a ∈ F that P extends to F(√a) if and only if P and finally that every ordering P of F extends to E if the degree of E over F is odd. | 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 | ordered fields | pl |
dc.subject | quadratic extensions | pl |
dc.subject | extensions of odd degree | pl |
dc.title | Extensions of Orderings | 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-2023-0027 | - |
dc.description.Affiliation | Institute of Informatics, University of Gdańsk, Poland | pl |
dc.description.references | Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191–198, 2015. doi:10.1007/s10817-015-9345-1. | pl |
dc.description.references | Adam Grabowski, Artur Korniłowicz, and Christoph Schwarzweller. Equality in computer proof-assistants. In Ganzha, Maria and Maciaszek, Leszek and Paprzycki, Marcin, editor, Proceedings of the 2015 Federated Conference on Computer Science and Information Systems, volume 5 of ACSIS-Annals of Computer Science and Information Systems, pages 45–54. IEEE, 2015. doi:10.15439/2015F229. | 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 | Alexander Prestel. Lectures on Formally Real Fields. Springer-Verlag, 1984. | pl |
dc.description.references | Knut Radbruch. Algebra I. Lecture Notes, University of Kaiserslautern, Germany, 1991. | pl |
dc.description.references | Knut Radbruch. Geordnete Körper. Lecture Notes, University of Kaiserslautern, Germany, 1991. | pl |
dc.description.references | Piotr Rudnicki, Christoph Schwarzweller, and Andrzej Trybulec. Commutative algebra in the Mizar system. Journal of Symbolic Computation, 32(1/2):143–169, 2001. doi:10.1006/jsco.2001.0456. | 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. 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. Ordered rings and fields. Formalized Mathematics, 25(1):63–72, 2017. doi:10.1515/forma-2017-0006. | pl |
dc.description.references | Christoph Schwarzweller and Agnieszka Rowińska-Schwarzweller. Quadratic extensions. Formalized Mathematics, 29(4):229–240, 2021. doi:10.2478/forma-2021-0021. | pl |
dc.identifier.eissn | 1898-9934 | - |
dc.description.volume | 31 | pl |
dc.description.issue | 1 | pl |
dc.description.firstpage | 341 | pl |
dc.description.lastpage | 352 | pl |
dc.identifier.citation2 | Formalized Mathematics | pl |
dc.identifier.orcid | 0000-0001-9587-8737 | - |
Występuje w kolekcji(ach): | Formalized Mathematics, 2023, Volume 31, Issue 1 |
Pliki w tej pozycji:
Plik | Opis | Rozmiar | Format | |
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Extensions-of-Orderings.pdf | 297,02 kB | Adobe PDF | Otwórz |
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