Relationship between primitive roots and quadratic residues I understand that if $g$ is a primitive root modulo an odd prime $p$, then Euler's Criterion tells us that $g$ cannot be a quadratic residue.
My question is, does this result generalize to prime powers? That is, if $g$ is a primitive root modulo $p^m$ for an odd prime $p$, must $g$ be a quadratic non-residue?
I spent a little while trying to come up with a proof but was unable to achieve anything useful so any proofs (or counterexamples) are welcome.

If $g$ is a primitive root, then the quadratic residues are precisely those elements which are even powers of $g$, so that if $g$ is a primitive root, then it cannot be a quadratic residue (since $g$ is an odd power of $g$)
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