MATH Seminar

Title: The degrees of divisors of $x^n-1$
Seminar: Number Theory
Speaker: Lola Thompson of University of Georgia
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2013-02-06 at 3:00PM
Venue: W306
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Abstract: We discuss what is known about the following questions concerning the degrees of divisors of $x^n-1 in Z[x]$, as n ranges over the natural numbers:\\ \\ 1. How often does $x^n-1$ have AT LEAST ONE divisor of every degree between 1 and n?\\ \\ 2. How often does $x^n-1$ have AT MOST ONE divisor of every degree between 1 and n?\\ \\ 3. How often does $x^n-1$ have EXACTLY ONE divisor of every degree between 1 and n?\\ \\ 4. For a given m, how often does $x^n-1$ have a divisor of degree m?\\ \\ We will also discuss what changes when Z is replaced by the finite field $F_p$. A portion of this talk is based on joint work with Paul Pollack.