Higher residue symbols

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 two different ways to compute this degree. The first method is using ramifiaction theory. The second proof follwos from our study of the distribution of primes $p$ for which all of $a_i$ are $l^{th}$ power residue simultaneously.