Euler’s criterion for eleventh power nonresidues

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 $$D=\displaystyle {2,7,11}$$ .