Little Bezout Theorem (Factor Theorem)

Journal of Formalized Mathematics
Volume 15, 2003
University of Bialystok
Copyright (c) 2003 Association of Mizar Users

Little Bezout Theorem (Factor Theorem)

Piotr Rudnicki: University of Alberta, Edmonton, Canada

Summary.

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.

MML Identifier: UPROOTS

The terminology and notation used in this paper have been introduced in the following articles [27] [37] [31] [8] [2] [26] [32] [15] [20] [38] [6] [7] [3] [9] [36] [33] [24] [23] [11] [21] [16] [19] [17] [18] [1] [12] [34] [28] [22] [10] [35] [4] [25] [39] [13] [29] [14] [30] [5]

Contents (PDF format)

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

Bibliography

[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.

Received December 30, 2003

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