Abstract
Let p be a prime \(\equiv 1 \!\!\!\pmod {11}\). If an integer D with \((p,D)=1\) is an eleventh power nonresidue\(\pmod {p}\), then \(D^{(p-1)/11} \equiv \alpha \, \!\!\!\pmod {p}\) for some eleventh root of unity \(\alpha (\not \equiv 1)\,\!\!\!\pmod {p}\). In this paper, we establish an explicit expression for \(\alpha \) in terms of a particular solution of certain quadratic partition of p. Euler’s criterion for eleventh power residues and nonresidues is given with explicit results for \(D=\displaystyle {2,7,11}\).
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Acknowledgements
The first author would like to thank the Council of Scientific and Industrial Research (CSIR), New Delhi, India for financial support during his research study at Bhaskaracharya Institute of Mathematics, Pune, India. He would also like to express his gratitude towards Central University of Jharkhand, Ranchi, India where the paper was finalized.
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Katre, S.A., Tanti, J. Euler’s criterion for eleventh power nonresidues. Proc Math Sci 129, 41 (2019). https://doi.org/10.1007/s12044-019-0482-z
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DOI: https://doi.org/10.1007/s12044-019-0482-z