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Nagell proved that for each prime $p\equiv 1\pmod{3}$, $p > 7$, there is aprime $q<2p^{1/2}$ that is a cubic residue modulo $p$. Here we show that foreach fixed $\epsilon > 0$, and each prime $p\equiv 1\pmod{3}$ with $p >p_0(\epsilon)$, the number of prime cubic residues $q < p^{1/2+\epsilon}$exceeds $p^{\epsilon/30}$. Our argument, like Nagell's, is rooted in the law ofcubic reciprocity; somewhat surprisingly, character sum estimates play no role.We use the same method to establish related results about prime quadratic andbiquadratic residues. For example, for all large primes $p$, there are morethan $p^{1/9}$ prime quadratic residues $q<p$.

Small prime $k$th power residues for $k=2,3,4$: A reciprocity laws approach