1. bookVolume 29 (2021): Issue 4 (December 2021)
Journal Details
License
Format
Journal
eISSN
1898-9934
First Published
09 Jun 2008
Publication timeframe
4 times per year
Languages
English
access type Open Access

Quadratic Extensions

Published Online: 09 Jul 2022
Volume & Issue: Volume 29 (2021) - Issue 4 (December 2021)
Page range: 229 - 240
Accepted: 30 Nov 2021
Journal Details
License
Format
Journal
eISSN
1898-9934
First Published
09 Jun 2008
Publication timeframe
4 times per year
Languages
English
Summary

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 aF such that E and ( Fa F\sqrt a ) are isomorphic over F.

Keywords

MSC

[1] 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. Open DOISearch in Google Scholar

[2] 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/s10817-017-9440-6.604425130069070 Open DOISearch in Google Scholar

[3] 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. Open DOISearch in Google Scholar

[4] Nathan Jacobson. Basic Algebra I. Dover Books on Mathematics, 1985. Search in Google Scholar

[5] Serge Lang. Algebra. Springer Verlag, 2002 (Revised Third Edition). Search in Google Scholar

[6] Heinz Lüneburg. Gruppen, Ringe, Körper: Die grundlegenden Strukturen der Algebra. Oldenbourg Verlag, 1999.10.1524/9783486599022 Search in Google Scholar

[7] Knut Radbruch. Algebra I. Lecture Notes, University of Kaiserslautern, Germany, 1991. Search in Google Scholar

[8] Christoph Schwarzweller. Ring and field adjunctions, algebraic elements and minimal polynomials. Formalized Mathematics, 28(3):251–261, 2020. doi:10.2478/forma-2020-0022. Open DOISearch in Google Scholar

[9] Christoph Schwarzweller. Formally real fields. Formalized Mathematics, 25(4):249–259, 2017. doi:10.1515/forma-2017-0024. Open DOISearch in Google Scholar

[10] Christoph Schwarzweller. On roots of polynomials and algebraically closed fields. Formalized Mathematics, 25(3):185–195, 2017. doi:10.1515/forma-2017-0018. Open DOISearch in Google Scholar

[11] Christoph Schwarzweller and Artur Korniłowicz. Characteristic of rings. Prime fields. Formalized Mathematics, 23(4):333–349, 2015. doi:10.1515/forma-2015-0027. Open DOISearch in Google Scholar

[12] Steven H. Weintraub. Galois Theory. Springer-Verlag, 2 edition, 2009.10.1007/978-0-387-87575-0 Search in Google Scholar

Recommended articles from Trend MD

Plan your remote conference with Sciendo