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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

4 Issues per year

SCImago Journal Rank (SJR) 2016: 0.207
Source Normalized Impact per Paper (SNIP) 2016: 0.315

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Volume 24, Issue 3


Some Algebraic Properties of Polynomial Rings

Christoph Schwarzweller / Artur Korniłowicz / Agnieszka Rowinska-Schwarzweller
Published Online: 2017-02-21 | DOI: https://doi.org/10.1515/forma-2016-0019


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


  • [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: CrossrefGoogle Scholar

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

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

About the article

Received: 2016-06-30

Published Online: 2017-02-21

Published in Print: 2016-09-01

Citation Information: Formalized Mathematics, Volume 24, Issue 3, Pages 227–237, ISSN (Online) 1898-9934, DOI: https://doi.org/10.1515/forma-2016-0019.

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

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Christoph Schwarzweller
Formalized Mathematics, 2017, Volume 25, Number 3

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