Prove or give a counterexample:
Every polynomial of degree 3 having coefficients over $\mathbb{F}_3$ has always a root in $\mathbb{F}_{27}$.
I noticed that $\mathbb{F}_{27} = \mathbb{F}_{3}[x]/(f)$, for f irreducible in $\mathbb{F}_{3}[x]$. I wrote down the form of an arbitrary polynomial of degree 3 over GF(27) and tried to find an irreducible polynomial (i.e. no roots), but this did not work out.
Appreciate your help!