On the degrees of divisors of T^n-1

Abstract

Fix a field $F$. In this paper, we study the sets \D_F(n) \subset [0,n] defined by [\D_F(n):= {0 \leq m \leq n: T^n-1\text{has a divisor of degree $m$ in} F[T]}.] When \D_F(n) consists of all integers $m$ with $0 \leq m \leq n$, so that $T^n-1$ has a divisor of every degree, we call $n$ an $F$-practical number. The terminology here is suggested by an analogy with the practical numbers of Srinivasan, which are numbers $n$ for which every integer $0 \leq m \leq \sigma(n)$ can be written as a sum of distinct divisors of $n$. Our first theorem states that, for any number field $F$ and any $x \geq 2$, [#{\text{$F$-practical $n\leq x$}} \asymp_{F} \frac{x}{\log{x}};] this extends work of the second author, who obtained this estimate when F=\Q. Suppose now that $x \geq 3$, and let $m$ be a natural number in $[3,x]$. We ask: For how many $n \leq x$ does $m$ belong to \D_F(n)? We prove upper bounds in this problem for both F=\Q and F=\F_p (with $p$ prime), the latter conditional on the Generalized Riemann Hypothesis. In both cases, we find that the number of such $n \leq x$ is $\ll_{F} x/(\log{m})^{2/35}$, uniformly in $m$