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.