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 $0\le m\le n$, 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 $0\le m\le \ddot{I}ƒ(n)$ can be written as a sum of distinct divisors of $n$. Our first theorem states that, for any number field $F$ and any $x\ge 2$, [#{\text{$F$-practical $n\le x$}} \asymp_{F} \frac{x}{\log{x}};] this extends work of the second author, who obtained this estimate when $F=\Q$. Suppose now that $x\ge 3$, and let $m$ be a natural number in $[3,x]$. We ask: For how many $n\le x$ 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 $n\le x$ is ${\ll }_{F}x/(\log m{)}^{2/35}$, uniformly in $m$.