Publishing house / Journals and Serials / Acta Arithmetica / All issues

Abstract

Let $ l, p$ be rational primes, $p\equiv 1 \pmod {l}$ and $\gamma $ a primitive root modulo $p$. If an integer $D$ with $(p,D)=1$ is an $l$th power nonresidue modulo $p$ then $D^{(p-1)/l}$ is an $l$th root of unity $\alpha \not \equiv 1 \pmod {p}$. Euler’s criterion for order $l$ modulo ${p}$ gives explicit conditions when $D^{(p-1)/l}\equiv \gamma ^{(p-1)/l}\pmod {p}$, i.e., $\operatorname{Ind} _\gamma D\equiv 1\pmod {l}$. We establish Euler’s criterion for order $l$ when the ring of integers in the cyclotomic field $\mathbb {Q}(\exp (2\pi i/l))$ of order $l$ is a PID. Conditions are obtained in terms of Jacobi sums of order $l$.