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\begin{document}$$\displaystyle l, p$$\end{document} be odd rational primes with p≡1(modl)\documentclass[12pt]{minimal}
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\begin{document}$$p\equiv 1 \!\!\!\pmod {l}$$\end{document} and γ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma $$\end{document} a primitive root (modp)\documentclass[12pt]{minimal}
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\begin{document}$$\pmod {p}$$\end{document}. An integer D with (p,D)=1\documentclass[12pt]{minimal}
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\begin{document}$$(p,D)=1$$\end{document}, is an l-th power residue or nonresidue (modp)\documentclass[12pt]{minimal}
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\begin{document}$$\pmod {p}$$\end{document} according to whether D(p-1)/l\documentclass[12pt]{minimal}
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\begin{document}$$D^{(p-1)/l}$$\end{document} is 1 or not, in which case it is an l-th root of unity (≠1)(modp)\documentclass[12pt]{minimal}
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\begin{document}$$(\ne 1)\,\pmod {p}$$\end{document}. 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≡γ(p-1)/l(modp)\documentclass[12pt]{minimal}
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\begin{document}$$D^{(p-1)/l}\equiv \gamma ^{(p-1)/l}\!\!\!\pmod {p}$$\end{document}, i.e., IndγD≡1(modl)\documentclass[12pt]{minimal}
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\begin{document}$$Ind_\gamma D\equiv 1\!\!\!\pmod {l}$$\end{document}. In this paper we establish the Euler’s criterion for orders l=23,29\documentclass[12pt]{minimal}
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\begin{document}$$l=23,\, 29$$\end{document} and 31, which are some least non-PID cases. Conditions are obtained in terms of Jacobi sums of respective orders.
