Polynomials with divisors of every degree

Abstract

We consider polynomials of the form t^n-1 and determine when members of this family have a divisor of every degree in Z[t]. With F(x) defined to be the number of such integers up to x, we prove the existence of two positive constants c_1 and c_2 such that $$c_{1}x/(logx)\le F(x)\le c_{2}x/(logx).$$

arXiv comment18 pages, to appear in the Journal of Number Theory