Let $F$ be a field of order 32. $0_F\neq a\in F$ and $a\neq1_F$. What is the set $\{a^n\,|\,n\geq1\}$?
I started by saying that $|F|=32=2^5$. Hence, $\text{char}(F)=2$ and obviously $\forall{a}\in F, a+a=0_F$. After this, I don't know how to continue to find that set. I heared that the answer is the set $F\setminus\{0_F\}$ that is of order 31 (I don't know if it's correct or not). But I don't know how to show that. I also considered that finite fields of the same order are isomorphic. Any help is so musch appreciated!
Note. Please consider that I recently started studying linear algebra. I'm reading K. Hoffman and R. Kunze book. I haven't studied abstract algebra and I don't know about rings, groups, group extensions, etc. I appreciate paying attention to this and giving any help that uses only field axioms, field properties, field characteristic and so on.