Some Algebraic Properties of Polynomial Rings

Christoph Schwarzweller 1 , Artur Korniłowicz 2  and Agnieszka Rowinska-Schwarzweller 3
  • 1 Institute of Computer Science University of Gdansk, Poland
  • 2 Institute of Informatics University of Białystok, Poland
  • 3 Sopot Poland


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.

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  • [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:

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

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


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