$x^3+2x+2 \in \mathbb{F}_3[x]$. 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 $\alpha$?

Careful, $\operatorname{GF}(27) \neq \mathbb Z/27\mathbb Z$ !
The latter is a ring of order $27$ which has zero-divisors ($3 \cdot 9 = 0$) and so not a field, and whose elements are $0$ up to $26$.
The first is a field of order $27$ of characteristic $3$, in which $\{0,1,\ldots, 26\} = \{0, 1, 2\}$ because $4=1$ etc.
