On Euler’s criterion for cubic nonresidues

Williams, Kenneth S.

doi:10.1090/S0002-9939-1975-0366792-0

On Euler’s criterion for cubic nonresidues
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by Kenneth S. Williams

Proc. Amer. Math. Soc. 49 (1975), 277-283

DOI: https://doi.org/10.1090/S0002-9939-1975-0366792-0

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Abstract:

If $p$ is a prime $\equiv 1 \pmod 3$ there are integers $L$ and $M$ such that $4p = {L^2} + 27{M^2},L \equiv 1\pmod 3$. Indeed $L$ and ${M^2}$ are unique. If $D$ is a cubic nonresidue $\pmod p$ it is shown how to choose the sign of $M$ so that \[ {D^{(p - 1)/3}} \equiv (L + 9M)/(L - 9M)\pmod p.\] The case $D = 2$ has been treated by Emma Lehmer.

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