Cyclotomic polynomials and nonstandard dice

Cyclotomic Polynomials

If n is a positive integer, then the n cyclotomic polynomial is defined as the unique monic polynomial having exactly the primitive n roots of unity as its zeros. In this paper we start off by examining some of the properties of cyclotomic polynomials; specifically focusing on their irreducibility and how they relate to primes. After that we explore some applications of these polynomials, inclu...

Cyclotomic Polynomials

Rolling Dice with Polynomials

George Sichermann in 1978 discovered a way to label two six-sided dice that produces outcomes identical to the one which occurs when two standard six-sided dice are rolled. In this talk, I will describe a generalization of this result due to Gallian and Rusin which uses polynomial algebra. I will conclude with a description of several variations on the original problem, several open questions, ...

The cyclotomic polynomials

1. The definition and general results We use the notation e(t) = e 2πit. Note that e(n) = 1 for integers n, e(1 2) = −1 and e(s + t) = e(s)e(t) for all s, t. Consider the polynomial x n − 1. The complex factorisation is obvious: the zeros of the polynomial are e(k/n) for 1 ≤ k ≤ n, so x n − 1 = n k=1 x − e k n. (1) One of these factors is x − 1, and when n is even, another is x + 1. The remaini...

Ducci-sequences and cyclotomic polynomials

We study Ducci-sequences using basic properties of cyclotomic polynomials over F2. We determine the period of a given Ducci-sequence in terms of the order of a polynomial, and in terms of the multiplicative orders of certain elements in finite fields. We also compute some examples and study links between Duccisequences, primitive polynomials and Artin’s conjecture on primitive roots. © 2005 Els...