Primitive Roots of Unity and Cyclotomic Polynomials

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

Primitive Roots of Unity and Cyclotomic Polynomials

Broderic Arneson: University of Alberta, Edmonton, Canada
Piotr Rudnicki: University of Alberta, Edmonton, Canada

Summary.

We present a formalization of roots of unity, define cyclotomic polynomials and demonstrate the relationship between cyclotomic polynomials and unital polynomials.

This work has been supported by NSERC Grant OGP9207.

MML Identifier: UNIROOTS

The terminology and notation used in this paper have been introduced in the following articles [33] [42] [34] [13] [9] [15] [35] [18] [2] [27] [36] [17] [25] [5] [43] [6] [7] [4] [16] [11] [40] [37] [8] [10] [28] [12] [26] [19] [20] [23] [21] [22] [24] [1] [41] [44] [29] [14] [38] [32] [3] [39] [31] [45] [30]

Contents (PDF format)

Preliminaries
Multiplicative Group of a Skew Field
Roots of Unity
The Unital Polynomial $x^n - 1$
Cyclotomic Polynomials

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. The ordinal numbers. Journal of Formalized Mathematics, 1, 1989.

[4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Journal of Formalized Mathematics, 1, 1989.

[5] Czeslaw Bylinski. Binary operations. 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. Partial functions. Journal of Formalized Mathematics, 1, 1989.

[9] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.

[10] Czeslaw Bylinski. The complex numbers. Journal of Formalized Mathematics, 2, 1990.

[11] Czeslaw Bylinski. Finite sequences and tuples of elements of a non-empty sets. Journal of Formalized Mathematics, 2, 1990.

[12] Czeslaw Bylinski. The sum and product of finite sequences of real numbers. Journal of Formalized Mathematics, 2, 1990.

[13] Library Committee. Introduction to arithmetic. Journal of Formalized Mathematics, Addenda, 2003.

[14] Agata Darmochwal. Finite sets. Journal of Formalized Mathematics, 1, 1989.

[15] Krzysztof Hryniewiecki. Basic properties of real numbers. Journal of Formalized Mathematics, 1, 1989.

[16] Andrzej Kondracki. Basic properties of rational numbers. Journal of Formalized Mathematics, 2, 1990.

[17] Eugeniusz Kusak, Wojciech Leonczuk, and Michal Muzalewski. Abelian groups, fields and vector spaces. Journal of Formalized Mathematics, 1, 1989.

[18] Rafal Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Journal of Formalized Mathematics, 2, 1990.

[19] Anna Justyna Milewska. The field of complex numbers. Journal of Formalized Mathematics, 12, 2000.

[20] Anna Justyna Milewska. The Hahn Banach theorem in the vector space over the field of complex numbers. Journal of Formalized Mathematics, 12, 2000.

[21] Robert Milewski. The evaluation of polynomials. Journal of Formalized Mathematics, 12, 2000.

[22] Robert Milewski. Fundamental theorem of algebra. Journal of Formalized Mathematics, 12, 2000.

[23] Robert Milewski. The ring of polynomials. Journal of Formalized Mathematics, 12, 2000.

[24] Robert Milewski. Trigonometric form of complex numbers. Journal of Formalized Mathematics, 12, 2000.

[25] Michal Muzalewski. Construction of rings and left-, right-, and bi-modules over a ring. Journal of Formalized Mathematics, 2, 1990.

[26] 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.

[27] Jan Popiolek. Some properties of functions modul and signum. Journal of Formalized Mathematics, 1, 1989.

[28] Jan Popiolek. Real normed space. Journal of Formalized Mathematics, 2, 1990.

[29] Konrad Raczkowski. Integer and rational exponents. Journal of Formalized Mathematics, 2, 1990.

[30] Piotr Rudnicki. Little Bezout theorem (factor theorem). Journal of Formalized Mathematics, 15, 2003.

[31] Piotr Rudnicki and Andrzej Trybulec. Multivariate polynomials with arbitrary number of variables. Journal of Formalized Mathematics, 11, 1999.

[32] Andrzej Trybulec. Enumerated sets. Journal of Formalized Mathematics, 1, 1989.

[33] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.

[34] Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.

[35] Michal J. Trybulec. Integers. Journal of Formalized Mathematics, 2, 1990.

[36] Wojciech A. Trybulec. Vectors in real linear space. Journal of Formalized Mathematics, 1, 1989.

[37] Wojciech A. Trybulec. Binary operations on finite sequences. Journal of Formalized Mathematics, 2, 1990.

[38] Wojciech A. Trybulec. Groups. Journal of Formalized Mathematics, 2, 1990.

[39] Wojciech A. Trybulec. Linear combinations in real linear space. Journal of Formalized Mathematics, 2, 1990.

[40] Wojciech A. Trybulec. Pigeon hole principle. Journal of Formalized Mathematics, 2, 1990.

[41] Wojciech A. Trybulec. Subgroup and cosets of subgroups. Journal of Formalized Mathematics, 2, 1990.

[42] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.

[43] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.

[44] Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle ratio. Journal of Formalized Mathematics, 10, 1998.

[45] 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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