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Euler’s criterion for prime order in the PID case

Volume 192 / 2020

Jagmohan Tanti Acta Arithmetica 192 (2020), 259-265 MSC: Primary 11T22, 11T24. DOI: 10.4064/aa180723-5-2 Published online: 13 November 2019


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 $l$.


  • Jagmohan TantiDepartment of Mathematics
    Central University of Jharkhand
    Ranchi 835205, India

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