Higher residue symbols

TL;DR

This paper determines the exact degree of certain radical extensions over the rationals involving $l^{th}$ roots of integers, providing two methods: one via ramification theory and another through prime distribution analysis.

Contribution

It introduces two novel approaches to compute the degree of radical extensions, enhancing understanding of their structure and prime distribution properties.

Findings

Exact degree formulas for radical extensions.

Two methods: ramification theory and prime distribution analysis.

Insights into prime behavior related to $l^{th}$ power residues.

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.