Higher Residue Symbol

TL;DR

This paper determines the exact degree of certain radical extensions over rationals, provides an algorithm for computation, and explores its implications for prime distribution and matrix rank related to $l^{th}$ power residues.

Contribution

It introduces a method to compute the degree of radical extensions and links it to prime distribution and matrix rank, simplifying the understanding of these extensions.

Findings

Exact degree computation algorithm for radical extensions.

Relation between extension degree and prime distribution for $l^{th}$ powers.

Connection between extension degree and the rank of an associated matrix.

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.