Consider a cyclic code of length $13$ over $\mathbb{F}_3$. Its cyclotomic cosets are $$C_0=\{0\},\,C_1=\{1,3,9\},\,C_2=\{2,6,5\},\,C_4=\{4,12,10\},\,C_5=\{7,8,11\}$$ and so there can only be $1,3,4,6,7,9,10,12$ (and $13$) dimensional cyclic codes of length $13$.
Now, I want to find a primitive $13$th-root of unity. I showed that $x^3+2x+1$ is irreducible over $\mathbb{F}_3$. This fact helps me find such primitive root of unity but I can't seem to figure this out.
Moreover, knowing the above, how can I say that $g(x)=x^7+2x^6+x^5+2x^4+x^2+2$ is a generator for a $[13,6,d]$ cyclic code with $d\geq 5$?
I appreciate any help on these two questions. Thanks in advance!