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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
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Page Count: 18
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Strong Divisibility, Cyclotomic Polynomials, and Iterated Polynomials
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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.

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