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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

4 Issues per year


SCImago Journal Rank (SJR) 2015: 0.134
Source Normalized Impact per Paper (SNIP) 2015: 0.686
Impact per Publication (IPP) 2015: 0.296

Open Access
Online
ISSN
1898-9934
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In This Section
Volume 24, Issue 3 (Sep 2016)

Issues

Some Algebraic Properties of Polynomial Rings

Christoph Schwarzweller
  • Corresponding author
  • Institute of Computer Science University of Gdansk, Poland
/ Artur Korniłowicz
  • Institute of Informatics University of Białystok, Poland
/ Agnieszka Rowinska-Schwarzweller
  • Sopot Poland
Published Online: 2017-02-21 | DOI: https://doi.org/10.1515/forma-2016-0019

Abstract

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]/<p> is isomorphic to the field of polynomials with degree smaller than the one of p.

MSC: 12E05; 11T55; 03B35

Keywords: polynomial; polynomial ring; polynomial GCD

MML: identifier: RING_4; version: 8.1.05 5.37.1275

References

  • [1] Grzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pak, 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: [Crossref] [Crossref]

  • [2] H. Heuser. Lehrbuch der Analysis. B.G. Teubner Stuttgart, 1990.

  • [3] Steven H. Weintraub. Galois Theory. Springer Verlag, 2 edition, 2009.

About the article

Received: 2016-06-30

Published Online: 2017-02-21

Published in Print: 2016-09-01



Citation Information: Formalized Mathematics, ISSN (Online) 1898-9934, DOI: https://doi.org/10.1515/forma-2016-0019. Export Citation

© by Christoph Schwarzweller. This work is licensed under version 3.0 of the Creative Commons Attribution–ShareAlike License (CC BY-SA 3.0 LEGALCODE)

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