How to find a primitive cube root of unity
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For any prime $p \ne 3$, we have $p^2 \equiv 1 \pmod 3$, and thus a primitive $3$rd root of unity exists in $\mathbf F_{p^2}$.
Assuming $p^2 \equiv 1 \pmod 3$, for any $x \in \mathbf F_{p^2}^*$, $x^\frac{p^2-1}3$ is a $3$rd root of unity, and moreover, all $3$rd roots of unity can be obtained in this manner.
Given a $3$rd root of unity, it is easy to check whether it is primitive.
If $x$ is chosen uniformly from $\mathbf F_{p^2}^*$, $x^\frac{p^2-1}3$ is a primitive $3$rd root of unity with probability $2/3$.