Power residue character of rational primes (Q2764521)

Let \(p\) and \(q\) be odd primes such that \(q \equiv 1 \bmod p\). It is a classical problem to characterize the primes \(r \neq q\) that are \(p\)th power residues modulo \(q\). Euler, for example, conjectured that \(2\) is a cubic residue modulo \(q\) if and only if \(q\) is represented by the quadratic form \(x^2 + 27y^2\). Criteria for fifth power residues involve systems of quadratic forms. In this article, the authors use Eisenstein's reciprocity law for rational integers and primary elements in the cyclotomic ring \(\mathbb Z[\zeta_p]\) to derive a general criterion, and then they give applications to cubic residuacity as well as to the \(p\)th power residue character of small primes for \(p = 5\), \(7\) and \(11\).