The Rabin's test says that $f$ is an irreductible polynomial over $\mathbb{F}_p$ if and only if
$f\mid (x^{p^{n}}-x)$
mcd$(f,x^{p^{n/q}}-x)=1$ $\forall q\mid n$
We are searching conditions for $p$ that satisfies the Rabin's test for the specific case on a $3^{rd}$ degree irreductible polynomial. So, we are searching conditions for $p$ that:
$f\mid (x^{p^{3}}-x)$
gcd$(f,x^{p}-x)=1$
We found that for example, with $f(x)=x^3-x+1$ and $p\in \{3, 13, 29, 31, 41, 47, 71\}$, $f(x)$ is irreductible in $\mathbb{F}_p$. But we don't know if that $p$'s have a characteritzation.