Mprime Seminar Series on Mathematics of Computer Algebra and Analysis
Strong Divisibility, Cyclotomic Polynomials, and Iterated Polynomials.
Jeff Sommars, Wheaton College
Abstract: In almost all algebra texts, one defines cyclotomic polynomials in reference to the primitive n'th roots of unity. However, as I will show, the property that gcd(x^n-1,x^m-1)=x^(gcd(m,n))-1 is sufficient to define the cyclotomic polynomials uniquely as polynomials in Z[x]. As applications, I will consider implications for cyclotomic polynomials, sequences of the form A^n-B^n, polynomial dynamical systems, and rigid divisibility sequences.