$x^3+2x+2 \in \mathbb{F}_3[x]$. Let $\alpha$ be a root in some extension field. One can see by brute force that $x^3+2x+2$ has no roots in GF(3). So it is irreducible and hence the minimal polynomial of $\alpha$.
My question is what is $\alpha$, how can I think about it? I determined that $GF(3)(\alpha) \cong GF(3^3)=GF(27)$. I then wrote a script to check $1,2,3,\dots,26$ to see if they are a root $\mod 3$ and $\mod 27$, and none of those are... Do I have a misconception about $\alpha$?

$GF(27)$ is the set of all elements of the form $pa^2+qa +r$, where $p,q,r \in \mathbb{Z_3}$. Thus accounting for 27 elements. With the additional constraint/relation that $a^3+2a^2+2=0$.
You may want to think of it as working with the expressions of the form $pa^2+qa +r$ modulo $3$ as well as modulo $a^3+2a^2+2$.
