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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On Euler's criterion for cubic nonresidues


Author: Kenneth S. Williams
Journal: Proc. Amer. Math. Soc. 49 (1975), 277-283
MSC: Primary 10A10
MathSciNet review: 0366792
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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

$\displaystyle {D^{(p - 1)/3}} \equiv (L + 9M)/(L - 9M)\pmod p.$

The case $ D = 2$ has been treated by Emma Lehmer.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0366792-0
PII: S 0002-9939(1975)0366792-0
Keywords: Euler's criterion, cyclotomy, cyclotomic numbers, root of unity modulo $ p$, cubic residues and nonresidues
Article copyright: © Copyright 1975 American Mathematical Society