I have the extension $\mathbb F_3[A]/\mathbb F_3$ where the $A$ are the roots of $x^{80}-1$.
I need to prove this extension is Galois, find the Galois group, and describe the automorphisms. but I'm having no success at all. the only thing I can think of is to look at $x^{81}-x$ and use the fact $x^{p^m}-x\mid x^{p^n}-x\iff m\mid n$ and $x^3-x\equiv 0$ somehow, but again, I don't know how to proceed.. :(
You're nearly there. You should know that the roots of $x^{81} - x$ form a finite field $E$ of order $81 = 3^{4}$, that any finite field is Galois over any of its subfields, and that the Galois group of $E$ over $\mathbb{F}_{3}$ is cyclic of order $4$, generated by the Frobenius automorphism $t \mapsto t^{3}$.