Journal of Formalized Mathematics
Volume 13, 2001
University of Bialystok
Copyright (c) 2001 Association of Mizar Users

On Polynomials with Coefficients in a Ring of Polynomials


Barbara Dzienis
University of Bialystok

Summary.

The main result of the paper is, that the ring of polynomials with $o_1$ variables and coefficients in the ring of polynomials with $o_2$ variables and coefficient in a ring $L$ is isomorphic with the ring with $o_1+o_2$ variables, and coefficients in $L$.

MML Identifier: POLYNOM6

The terminology and notation used in this paper have been introduced in the following articles [21] [27] [23] [13] [28] [8] [9] [20] [1] [22] [14] [24] [17] [11] [5] [10] [26] [12] [6] [2] [3] [4] [25] [7] [19] [15] [29] [18] [16]

Contents (PDF format)

  1. Preliminaries
  2. About Bags
  3. Main Results

Bibliography

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Journal of Formalized Mathematics, 1, 1989.
[2] Grzegorz Bancerek. The ordinal numbers. Journal of Formalized Mathematics, 1, 1989.
[3] Grzegorz Bancerek. Sequences of ordinal numbers. Journal of Formalized Mathematics, 1, 1989.
[4] Grzegorz Bancerek. Ordinal arithmetics. Journal of Formalized Mathematics, 2, 1990.
[5] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Journal of Formalized Mathematics, 1, 1989.
[6] Grzegorz Bancerek and Piotr Rudnicki. On defining functions on trees. Journal of Formalized Mathematics, 5, 1993.
[7] Jozef Bialas. Group and field definitions. Journal of Formalized Mathematics, 1, 1989.
[8] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[9] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
[10] Czeslaw Bylinski. Finite sequences and tuples of elements of a non-empty sets. Journal of Formalized Mathematics, 2, 1990.
[11] Agata Darmochwal. Finite sets. Journal of Formalized Mathematics, 1, 1989.
[12] Jaroslaw Kotowicz. Monotone real sequences. Subsequences. Journal of Formalized Mathematics, 1, 1989.
[13] Eugeniusz Kusak, Wojciech Leonczuk, and Michal Muzalewski. Abelian groups, fields and vector spaces. Journal of Formalized Mathematics, 1, 1989.
[14] Beata Madras. Product of family of universal algebras. Journal of Formalized Mathematics, 5, 1993.
[15] Robert Milewski. Associated matrix of linear map. Journal of Formalized Mathematics, 7, 1995.
[16] Robert Milewski. The ring of polynomials. Journal of Formalized Mathematics, 12, 2000.
[17] Michal Muzalewski. Construction of rings and left-, right-, and bi-modules over a ring. Journal of Formalized Mathematics, 2, 1990.
[18] Piotr Rudnicki and Andrzej Trybulec. Multivariate polynomials with arbitrary number of variables. Journal of Formalized Mathematics, 11, 1999.
[19] Christoph Schwarzweller. The field of quotients over an integral domain. Journal of Formalized Mathematics, 10, 1998.
[20] Andrzej Trybulec. Binary operations applied to functions. Journal of Formalized Mathematics, 1, 1989.
[21] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[22] Andrzej Trybulec. Many-sorted sets. Journal of Formalized Mathematics, 5, 1993.
[23] Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.
[24] Wojciech A. Trybulec. Vectors in real linear space. Journal of Formalized Mathematics, 1, 1989.
[25] Wojciech A. Trybulec. Groups. Journal of Formalized Mathematics, 2, 1990.
[26] Wojciech A. Trybulec. Pigeon hole principle. Journal of Formalized Mathematics, 2, 1990.
[27] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[28] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[29] Katarzyna Zawadzka. Sum and product of finite sequences of elements of a field. Journal of Formalized Mathematics, 4, 1992.

Received August 10, 2001


[ Download a postscript version, MML identifier index, Mizar home page]