Simple Extensions-Reference-Cited by-同舟云学术

Simple Extensions

Published:2023-09-01 Issue:1 Volume:31 Page:287-298
ISSN:1898-9934
Container-title:Formalized Mathematics
language:en
Short-container-title:

Author:

Schwarzweller Christoph¹^ORCID,Rowińska-Schwarzweller Agnieszka²

Affiliation:

1. 1 Institute of Informatics , University of Gdańsk , Poland

2. 2 Institute of Informatics , University of Gdańsk , Poland

Abstract

Summary 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].

Publisher

Walter de Gruyter GmbH

Subject

Applied Mathematics,Computational Mathematics

Link

https://www.sciendo.com/pdf/10.2478/forma-2023-0023

Reference15 articles.

1. Andreas Gathmann. Einführung in die Algebra. Lecture Notes, University of Kaiserslautern, Germany, 2011.

2. 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.

3. Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Mizar in a nutshell. Journal of Formalized Reasoning, 3(2):153–245, 2010.

4. 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.

5. Artur Korniłowicz. Flexary connectives in Mizar. Computer Languages, Systems & Structures, 44:238–250, December 2015. doi:10.1016/j.cl.2015.07.002.