Fix a field F. In this paper, we study the sets D_F(n) ⊂ [0,n] defined by D_F(n) := {0 ≦ m ≦ n : Tⁿ-1 has a divisor of degree m in F[T]}. When D_F(n) consists of all integers m with 0 ≦ m ≦ n, so that Tⁿ-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 ≦ m ≦ σ(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 ≧ 2, #{F-practical n≦ x} \asymp_F (x/log x); this extends work of the second author, who obtained this estimate when F=Q.

Suppose now that x ≧ 3, and let m be a natural number in [3,x]. We ask: For how many n ≦ 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 ≦ x is O_F (x/(log m)^2/35), uniformly in m.