You are not currently logged in.

Access JSTOR through your library or other institution:


Log in through your institution.

Strong Divisibility, Cyclotomic Polynomials, and Iterated Polynomials

Nathan Bliss, Ben Fulan, Stephen Lovett and Jeff Sommars
The American Mathematical Monthly
Vol. 120, No. 6 (June–July 2013), pp. 519-536
DOI: 10.4169/amer.math.monthly.120.06.519
Stable URL:
Page Count: 18
  • Download ($19.00)
  • Subscribe ($19.50)
  • Cite this Item
Strong Divisibility, Cyclotomic Polynomials, and Iterated Polynomials
Preview not available


Abstract Almost all algebra texts define cyclotomic polynomials using primitive nth roots of unity. However, the elementary formula \documentclass{article} \pagestyle{empty}\begin{document} $\gcd(x^m -1, x^n-1) = x^{\gcd(m,n)}-1$ \end{document} 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.

Page Thumbnails