Volume 15, 2003

University of Bialystok

Copyright (c) 2003 Association of Mizar Users

**Piotr Rudnicki**- University of Alberta, Edmonton, Canada

- We present a formalization of the factor theorem for univariate polynomials, also called the (little) Bezout theorem: Let $r$ belong to a commutative ring $L$ and $p(x)$ be a polynomial over $L$. Then $x-r$ divides $p(x)$ iff $p(r) = 0$. We also prove some consequences of this theorem like that any non zero polynomial of degree $n$ over an algebraically closed integral domain has $n$ (non necessarily distinct) roots.

This work has been supported by NSERC Grant OGP9207.

- Preliminaries
- Canonical Ordering of a Finite Set
- More about Bags
- More on Polynomials
- Little Bezout Theorem
- Polynomials Defined by Roots

- [1]
Grzegorz Bancerek.
Cardinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [2]
Grzegorz Bancerek.
The fundamental properties of natural numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [3]
Grzegorz Bancerek and Krzysztof Hryniewiecki.
Segments of natural numbers and finite sequences.
*Journal of Formalized Mathematics*, 1, 1989. - [4]
Grzegorz Bancerek and Piotr Rudnicki.
On defining functions on trees.
*Journal of Formalized Mathematics*, 5, 1993. - [5]
Jozef Bialas.
Group and field definitions.
*Journal of Formalized Mathematics*, 1, 1989. - [6]
Czeslaw Bylinski.
Functions and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [7]
Czeslaw Bylinski.
Functions from a set to a set.
*Journal of Formalized Mathematics*, 1, 1989. - [8]
Czeslaw Bylinski.
Some basic properties of sets.
*Journal of Formalized Mathematics*, 1, 1989. - [9]
Czeslaw Bylinski.
Finite sequences and tuples of elements of a non-empty sets.
*Journal of Formalized Mathematics*, 2, 1990. - [10]
Czeslaw Bylinski.
The modification of a function by a function and the iteration of the composition of a function.
*Journal of Formalized Mathematics*, 2, 1990. - [11]
Czeslaw Bylinski.
The sum and product of finite sequences of real numbers.
*Journal of Formalized Mathematics*, 2, 1990. - [12]
Agata Darmochwal.
Finite sets.
*Journal of Formalized Mathematics*, 1, 1989. - [13]
Andrzej Kondracki.
The Chinese Remainder Theorem.
*Journal of Formalized Mathematics*, 9, 1997. - [14]
Jaroslaw Kotowicz.
Monotone real sequences. Subsequences.
*Journal of Formalized Mathematics*, 1, 1989. - [15]
Eugeniusz Kusak, Wojciech Leonczuk, and Michal Muzalewski.
Abelian groups, fields and vector spaces.
*Journal of Formalized Mathematics*, 1, 1989. - [16]
Anna Justyna Milewska.
The field of complex numbers.
*Journal of Formalized Mathematics*, 12, 2000. - [17]
Robert Milewski.
The evaluation of polynomials.
*Journal of Formalized Mathematics*, 12, 2000. - [18]
Robert Milewski.
Fundamental theorem of algebra.
*Journal of Formalized Mathematics*, 12, 2000. - [19]
Robert Milewski.
The ring of polynomials.
*Journal of Formalized Mathematics*, 12, 2000. - [20]
Michal Muzalewski.
Construction of rings and left-, right-, and bi-modules over a ring.
*Journal of Formalized Mathematics*, 2, 1990. - [21]
Michal Muzalewski and Leslaw W. Szczerba.
Construction of finite sequence over ring and left-, right-, and bi-modules over a ring.
*Journal of Formalized Mathematics*, 2, 1990. - [22]
Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec, and Pauline N. Kawamoto.
Preliminaries to circuits, I.
*Journal of Formalized Mathematics*, 6, 1994. - [23]
Takaya Nishiyama and Yasuho Mizuhara.
Binary arithmetics.
*Journal of Formalized Mathematics*, 5, 1993. - [24]
Jan Popiolek.
Real normed space.
*Journal of Formalized Mathematics*, 2, 1990. - [25]
Piotr Rudnicki and Andrzej Trybulec.
Multivariate polynomials with arbitrary number of variables.
*Journal of Formalized Mathematics*, 11, 1999. - [26]
Andrzej Trybulec.
Semilattice operations on finite subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [27]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [28]
Andrzej Trybulec.
Tuples, projections and Cartesian products.
*Journal of Formalized Mathematics*, 1, 1989. - [29]
Andrzej Trybulec.
Many-sorted sets.
*Journal of Formalized Mathematics*, 5, 1993. - [30]
Andrzej Trybulec.
On the sets inhabited by numbers.
*Journal of Formalized Mathematics*, 15, 2003. - [31]
Andrzej Trybulec.
Subsets of real numbers.
*Journal of Formalized Mathematics*, Addenda, 2003. - [32]
Wojciech A. Trybulec.
Vectors in real linear space.
*Journal of Formalized Mathematics*, 1, 1989. - [33]
Wojciech A. Trybulec.
Binary operations on finite sequences.
*Journal of Formalized Mathematics*, 2, 1990. - [34]
Wojciech A. Trybulec.
Groups.
*Journal of Formalized Mathematics*, 2, 1990. - [35]
Wojciech A. Trybulec.
Linear combinations in real linear space.
*Journal of Formalized Mathematics*, 2, 1990. - [36]
Wojciech A. Trybulec.
Pigeon hole principle.
*Journal of Formalized Mathematics*, 2, 1990. - [37]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [38]
Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [39]
Katarzyna Zawadzka.
Sum and product of finite sequences of elements of a field.
*Journal of Formalized Mathematics*, 4, 1992.

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