Higher Residue Symbol

Abstract

Given a prime number $l$ and a finite set of integers $S=\{a_1,...,a_m\}$ we find out the exact degree of the extension $\mathbb{Q}(a_1^{\frac{1}{l}},...,a_m^{\frac{1}{l}})/\mathbb{Q}$. We give an algorithm to compute this degree and then further relate it to the study of the distribution of primes $p$ for which all of $a_i$ assume a preassigned $l^{th}$ power residue simultaneously. Also we relate this degree to rank of a matrix obtained from $S=\{a_1,...,a_m\}$. This latter arguement enable one to describe the degree $\mathbb{Q}(a_1^{\frac{1}{l}},...,a_m^{\frac{1}{l}})/\mathbb{Q}$ in much simpler terms.