Relationship between primitive roots and quadratic residues

Question

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.

Answer 1

The order of a quadratic residue modulo $n$ divides $\varphi(n)/2$. A primitive root has order $\varphi(n)$. Hence a primitive root is always a quadratic nonresidue.

Answer 2

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 $g$) .