Density of primes in l-th power residues

Abstract

Given a prime number l, a finite set of integers S = {a 1, ...,a m } and m many l-th roots of unity $\zeta_l^{r_i}, i=1, \ldots ,m$ we study the distribution of primes p in ℚ(ζ l ) such that the l-th residue symbol of a i with respect to p is $\zeta_l^{r_i}, \mbox{ for all } i$ . We find out that this is related to the degree of the extension $$\mathbb{Q}(a_1^{\frac{1}{l}}, \ldots ,a_m^{\frac{1}{l}})/\mathbb{Q}$$ . We give an algorithm to compute this degree. Also we relate this degree to rank of a matrix obtained from S = {a 1, ...,a m }. This latter argument enables one to describe the degree $$\mathbb{Q}(a_1^{\frac{1}{l}}, \ldots ,a_m^{\frac{1}{l}})/\mathbb{Q}$$ in much simpler terms.