• S A KATRE

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    • Euler’s criterion for eleventh power nonresidues

      S A KATRE JAGMOHAN TANTI

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      Let $p$ be a prime $\equiv 1\pmod{p}$. 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 $D=2, 7,11$.

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