Let \(\displaystyle l, p\) be odd rational primes with \(p\equiv 1 \!\!\!\pmod {l}\) and \(\gamma \) a primitive root \(\pmod {p}\) . An integer D with \((p,D)=1\) , is an l-th power residue or nonresidue \(\pmod {p}\) according to whether \(D^{(p-1)/l}\) is 1 or not, in which case it is an l-th root of unity \((\ne 1)\,\pmod {p}\) . In the case of D a non-residue, Euler’s criterion for order l aims to give the explicit conditions when \(D^{(p-1)/l}\equiv \gamma ^{(p-1)/l}\!\!\!\pmod {p}\) , i.e., \(Ind_\gamma D\equiv 1\!\!\!\pmod {l}\) . In this paper we establish the Euler’s criterion for orders \(l=23,\, 29\) and 31, which are some least non-PID cases. Conditions are obtained in terms of Jacobi sums of respective orders.
