(a computer assisted approach)

Formalized Mathematics

<abstract><title style='display:none'>Abstract</title> In this article we extend the algebraic theory of polynomial rings, formalized in Mizar [1], based on [2], [3]. After introducing constant and monic polynomials we present the canonical embedding of R into R[X] and deal with both unit and irreducible elements. We also define polynomial GCDs and show that for fields F and irreducible polynomials p the field F[X]/&lt;p&gt; is isomorphic to the field of polynomials with degree smaller than the one of p.</abstract>

Abstract

Institute of Computer Science University of Gdansk,

Institute of Informatics University of Białystok,

Keywords

<article-title>Some Algebraic Properties of Polynomial Rings</article-title>

Some Algebraic Properties of Polynomial Rings

{"article-title":"Some Algebraic Properties of Polynomial Rings"}

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<DIV align=left> <UL> <LI>Pioneering Journal in the field of formalized and mechanically checked mathematics.</LI> <LI>Excellent articles on latest research in formal methods and computer-checked mathematics.</LI> <LI>Full-length original research articles in English.</LI> <LI>Archival issues available at the journal's home page <A href="http://fm.mizar.org/">fm.mizar.org/</A>.</LI></UL> <UL> <LI>To build a uniform and extensible computer-checked repository of mathematical knowledge. </LI> <LI>Reconstruction of mathematics based on mechanization of mathematical reasoning. </LI> <LI>To check a wide range of mathematical theorems using a computer in an automatic fashion. </LI> <LI>General mathematics: especially topology, lattice theory, functional analysis. </LI> <LI>Applications: computer-aided verification of software and hardware, computer-aided teaching of mathematics. </LI></UL> Why subscribe and read <UL> <LI>Pioneering source of high quality research in formalized and mechanically checked mathematics. </LI> <LI>The oldest journal in the field, a medium for the publication of research reporting pioneering findings of broad mathematical significance that are likely to advance the direction and thinking of the field. </LI> <LI>Long-term experience in publishing original articles on mechanizing mathematics and various aspects of automated reasoning, since 1990. </LI></UL> Why submit <UL> <LI>Fair and constructive peer review by international experts in the field (three referees are designated for the assessment of each manuscript) - double-blind review process. </LI> <LI>New publishing technology: articles automatically translated from a formal language into English and automatically typeset, available in both forms - electronic and printed. </LI> <LI>Articles abstracted by online indexing services and databases. </LI></UL> Rejection Rate <UL> <LI>59% of submitted articles. </LI></UL> The rejection percentage is described in detail in the following paper, pp.239-240: <A href="http://dx.doi.org/10.1007/978-3-540-73086-6_20">dx.doi.org/10.1007/978-3-540-73086-6_20</A> Archiving Sciendo archives the contents of this journal in <A href="https://www.portico.org/">Portico</A> - digital long-term preservation service of scholarly books, journals and collections. Plagiarism Policy The editorial board is participating in a growing community of <A href="https://www.crossref.org/services/similarity-check/">Similarity Check System's</A> users in order to ensure that the content published is original and trustworthy. Similarity Check is a medium that allows for comprehensive manuscripts screening, aimed to eliminate plagiarism and provide a high standard and quality peer-review process. Back Issues <A href="http://fm.mizar.org/">fm.mizar.org/</A> Other Files <A href="http://fm.mizar.org/addenda/addenda2008.pdf">Addenda 2008</A> Formalized Mathematics <A href="http://fm.mizar.org/fm.bib">Bibliography</A> File. Formalized Mathematics <A href="http://fm.mizar.org/external.bib">External Bibliography</A> File. "In years: 2016-2017, the English version of Formalized Mathematics was financed under agreements 548/P-DUN/2016 and 548/1/P-DUN/2016 with the funds from the Polish Minister of Science and Higher Education for the dissemination of science." </DIV>

Editor-in-Chief <A href="http://mizar.org/people/romat/">Roman Matuszewski</A>, University of Bialystok, Poland Language Editor <A href="http://soar-rd.shinshu-u.ac.jp/profile/en.ypSpbpkh.html">Pauline N. Kawamoto</A>, Shinshu University, Japan Senior Advisory Board <UL> <LI><A href="http://en.wikipedia.org/wiki/William_Ward_Armstrong">William W. Armstrong</A>, University of Alberta, Canada, </LI> <LI><A href="https://www.cs.utexas.edu/users/boyer/">Robert S. Boyer</A>, University of Texas at Austin, USA, </LI> <LI><A href="http://www.macs.hw.ac.uk/staff-directory/fairouz-kamareddine.htm">Fairouz Kamareddine</A>, Heriot-Watt University, UK, </LI></UL> Editorial Board <UL> <LI><A href="http://www-csli.stanford.edu/people/alama-jesse">Jesse Alama</A>, Stanford University, USA, Thematic Editor - Logic </LI> <LI><A href="http://www.auburn.edu/~kuperkm/">Krystyna Kuperberg</A>, Auburn University, USA, Thematic Editor - Mathematics </LI> <LI><A href="http://www.cl.cam.ac.uk/~jrh13/">John Harrison</A>, Amazon Web Services, Portland, USA, Thematic Editor - Computer Science </LI> <LI><A href="http://andrej.com/">Andrej Bauer,</A> University of Ljubljana, Slovenia </LI> <LI><A href="http://math.uwb.edu.pl/~bylinski/">Czeslaw Bylinski</A>, University of Bialystok, Poland, </LI> <LI><A href="http://www.informatik.uni-trier.de/~ley/pers/hd/e/Endou:Noboru.html">Noboru Endou</A>, Gifu National College of Technology, Japan, </LI> <LI><A href="http://www.olsonet.com/pg/">Pawel Gburzynski</A>, Olsonet Communications Corporation, Canada, </LI> <LI><A href="http://math.uwb.edu.pl/~adam/">Adam Grabowski</A>, University of Bialystok, Poland, </LI> <LI><A href="http://www.diku.dk/~grue/">Klaus Grue</A>, University of Copenhagen, Denmark, </LI> <LI><A href="http://soar-rd.shinshu-u.ac.jp/profile/en.ypSpbpkh.html">Pauline N. Kawamoto</A>, Shinshu University, Japan, </LI> <LI><A href="https://kwarc.info/people/mkohlhase/">Michael Kohlhase</A>, Friedrich-Alexander-University, Erlangen-Nuremberg, Germany, </LI> <LI><A href="http://math.uwb.edu.pl/~arturk/index-en.html">Artur Kornilowicz</A>, University of Bialystok, Poland, </LI> <LI><A href="http://people.cas.uab.edu/~jcmayer/">John Mayer</A>, University of Alabama at Birmingham, USA, </LI> <LI><A href="http://math.uwb.edu.pl/~adamn/">Adam Naumowicz</A>, University of Bialystok, Poland, </LI> <LI><A href="http://scholar.google.com/scholar?q=Alexander+Shibakov&btnG=&hl=en&as_sdt=0%2C5">Alexander Shibakov</A>, Tennessee Tech University, USA, </LI> <LI><A href="http://soar-rd.shinshu-u.ac.jp/profile/en.OeceZVkh.html">Yasunari Shidama</A>, Shinshu University, Japan, </LI> <LI><A href="http://www.cs.miami.edu/~geoff/">Geoff Sutcliffe</A>, University of Miami, USA </LI></UL> Contact dr Roman Matuszewski University of Bialystok Institute of Informatics ul. Ciolkowskiego 1 M, room 2021 15-245 Białystok Poland mobile: +48 603 491 986 e-mail: <A href="mailto:romat@mizar.org">romat@mizar.org</A> Publisher De Gruyter Poland Bogumiła Zuga 32A Str. 01-811 Warsaw, Poland T: +48 22 701 50 15

Formalized Mathematics is issued quarterly and publishes papers which are abstracts of Mizar articles contributed to the <A href="http://mizar.org/library/">Mizar Mathematical Library</A> (MML). The main version of the Journal is the electronic version. The official abbreviation of the Journal is: Formaliz. Math. List of Reviewers <UL> <LI><A href="http://dblp.uni-trier.de/pers/hd/a/Arai:Kenichi">Kenichi Arai</A>, Tokyo University of Science, Japan </LI> <LI><A href="http://math.uwb.edu.pl/~bylinski/">Czeslaw Bylinski</A>, University of Bialystok, Poland </LI> <LI><A href="https://orcid.org/0000-0002-4901-0766">Marco Caminati</A>, University of St. Andrews, UK </LI> <LI><A href="https://orcid.org/0000-0002-4901-0766">Roland Coghetto</A>, La Louviere, Belgium </LI> <LI><A href="http://math.uwb.edu.pl/~adam/">Adam Grabowski</A>, University of Bialystok, Poland </LI> <LI><A href="https://dblp.uni-trier.de/pers/hd/i/Ivanov:Ievgen">Ievgen Ivanov</A>, Taras Shevchenko National University, Kyiv, Ukraine </LI> <LI><A href="https://dblp.uni-trier.de/pers/hd/j/Jaeger:Peter">Peter Jaeger</A>, Munich, Germany </LI> <LI><A href="http://cl-informatik.uibk.ac.at/users/cek/">Cezary Kaliszyk</A>, University of Innsbruck, Austria </LI> <LI><A href="http://math.uwb.edu.pl/~arturk/index-en.html">Artur Kornilowicz</A>, University of Bialystok, Poland </LI> <LI><A href="http://www.informatik.uni-trier.de/~ley/pers/hd/l/Liang:Xiquan.html">Xiquan Liang</A>, Qingdao University of Science and Technology, China </LI> <LI><A href="https://dblp.uni-trier.de/pers/hd/n/Nakamura:Yatsuka">Yatsuka Nakamura</A>, Shinshu University, Japan </LI> <LI><A href="http://math.uwb.edu.pl/~adamn/">Adam Naumowicz</A>, University of Bialystok, Poland </LI> <LI><A href="http://soar-rd.shinshu-u.ac.jp/profile/en.geDNbpkh.html">Hiroyuki Okazaki</A>, Shinshu University, Japan </LI> <LI><A href="http://alioth.uwb.edu.pl/~pakkarol/">Karol Pąk</A>, University of Bialystok, Poland </LI> <LI><A href="http://wiki.mizar.org/twiki/bin/view/Main/MarcoRiccardi">Marco Riccardi</A>, University of Pisa, Italy </LI> <LI><A href="https://inf.ug.edu.pl/~schwarzw/">Christoph Schwarzweller</A>, University of Gdansk, Poland </LI> <LI><A href="https://www.ciirc.cvut.cz/~urbanjo3/">Josef Urban</A>, Czech Institute of Informatics, Prague </LI> <LI><A href="http://soar-rd.shinshu-u.ac.jp/profile/en.gCnejaTN.html">Katsumi Wasaki</A>, Shinshu University, Japan </LI> <LI><A href="http://dblp.uni-trier.de/pers/hd/w/Watase:Yasushige">Yasushige Watase</A>, Tokyo, Japan </LI> <LI><A href="http://www.informatik.uni-trier.de/~ley/pers/hd/w/Wiedijk:Freek.html">Freek Wiedijk</A>, Radboud University, the Netherlands. </LI></UL> All papers are reviewed by at least three experts from the relevant field (double-blind review process). <A href="http://math.andrej.com/2013/08/19/how-to-review-formalized-mathematics/">How to review formalized mathematics</A> - guidelines by A. Bauer. The reviewing process is described in detail in the following paper, pp.239-240: <A href="http://dx.doi.org/10.1007/978-3-540-73086-6_20">dx.doi.org/10.1007/978-3-540-73086-6_20</A>, (<A href="https://inf.ug.edu.pl/~schwarzw/papers/mkm07.pdf">pdf</A> file) Journal Authorship Policy (ghost-writing and guest authorship) is <A href="http://fm.mizar.org/editorial-policy.html">here</A>. Mathematical Proof-Checked Journal - all papers are checked by the <A href="http://www.mizar.org/system/">Mizar System</A> and automatically translated into English. Open Access Statement The journal is an Open Access journal that allows a free unlimited access to all its contents without any restrictions upon publication to all users. Articles are available under the Creative Commons Attribution-ShareAlike License (CC-BY-SA) version 3.0 Unported (<A href="http://creativecommons.org/licenses/by-sa/3.0/">creativecommons.org/licenses/by-sa/3.0/</A>). A detailed report on choosing the license: <A href="http://arxiv.org/abs/1107.3212">arxiv.org/abs/1107.3212</A>. <A href="http://en.wikipedia.org/wiki/Open-access_journal">Open Access Policy</A>: this journal provides immediate open access to its content. The journal does not have article processing charges (APCs) nor article submission charges. <A href="https://fm.mizar.org/editorial-policy.html">Instruction for Authors</A> Articles should be submitted to: Adam Grabowski, Institute of Informatics University of Bialystok ul. Ciolkowskiego 1 M 15-245 Bialystok, Poland by e-mail: <A href="mailto:mml@mizar.uwb.edu.pl">mml@mizar.uwb.edu.pl</A>

Formalized Mathematics

Some Algebraic Properties of Polynomial Rings

Christoph Schwarzweller,
Christoph Schwarzweller,
Artur Korniłowicz and
Artur Korniłowicz and
Agnieszka Rowinska-Schwarzweller
Agnieszka Rowinska-Schwarzweller

Published Online: 21 Feb 2017

Volume & Issue: Volume 24 (2016) - Issue 3 (September 2016)

Page range: 227 - 237

Received: 30 Jun 2016

Journal & Issue Details

Formalized Mathematics

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References

Recent Articles

Formalized Mathematics

Some Algebraic Properties of Polynomial Rings

Christoph Schwarzweller,Christoph Schwarzweller,Artur Korniłowicz andArtur Korniłowicz andAgnieszka Rowinska-SchwarzwellerAgnieszka Rowinska-Schwarzweller

Published Online: 21 Feb 2017

Volume & Issue: Volume 24 (2016) - Issue 3 (September 2016)

Page range: 227 - 237

Received: 30 Jun 2016

Journal & Issue Details

Formalized Mathematics

PDF Preview

References

Recent Articles

Christoph Schwarzweller,
Christoph Schwarzweller,
Artur Korniłowicz and
Artur Korniłowicz and
Agnieszka Rowinska-Schwarzweller
Agnieszka Rowinska-Schwarzweller