  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 $m$.
