An application of higher reciprocity to computational number theory

Abstract

The Higher Reciprocity Laws are considered to be among the deepest and most fundamental results in number theory. Yet, they have until recently played no part in number theoretic algorithms. In this paper we explore the power of the laws in algorithms. The problem we consider is part of a group of well-studied problems about roots in finite fields and rings. Let F denote a finite field, let m denote a direct product of finite fields. Consider the following problems: Problem 1. Is Xn = a solvable in F; Problem 2. If Xn = a is solvable in F find X; Problem 3. Is Xn = a solvable in m; Problem 4. If Xn = a solvable in m find X.