I am trying to figure out the splitting field of $x^3 - x + 1$ over $\mathbb{F}_3$ .
It will be the field $\mathbb{F}_3[x]/\langle x^3 -x +1\rangle$ but I have to show irreducibility too.
All I know is that there is a rule that states that if a polynomial of degree $n$ over a finite field is irreducible if it is a factor of $x^{p^n -1} -1$. But in this case, I have to do long division of $x^{26} -1$. Is there any other way this can be done ?
Can mod-2 test be used here ?