Strong Divisibility, Cyclotomic Polynomials, and Iterated Polynomials

Abstract

Almost all algebra texts define cyclotomic polynomials using primitive nth roots of unity. However, the elementary formula gcd(xm - 1, xn - 1) = xgcd(m,n) - 1 in ℤ[x] can be used to define the cyclotomic polynomials without reference to roots of unity. In this article, partly motivated by cyclotomic polynomials, we prove a factorization property about strong divisibility sequences in a unique factorization domain. After illustrating this property with cyclotomic polynomials and sequences of the form An − Bn, we use the main theorem to prove that the dynamical analogues of cyclotomic polynomials over any unique factorization domain R are indeed polynomials in R[x]. Furthermore, a converse to this article's main theorem provides a simple necessary and sufficient condition for a divisibility sequence to be a rigid divisibility sequence.