Abstract
Let F(x) be a polynomial with coefficients in algebraic number field k. An estimate the number of irreducible cyclotomic factors of F in k[x] the number of irreducible noncyclotomic factors of F, the number of n-th roots of unity among the roots of F and the number of primitive n-th roots of unity among the roots of F are given by C. G. Pinner and J. D. Vaaler [8, 9]. All of these quantities are counted with multiplicity and estimated by expressions which depend explicitly on k, on the degree of F and height of F and on n. In this paper we shall motivate these two works of C. Pinner and J. Vaaler to give more sophisticated version and alternatively proof of this estimate.