Let $F$ be a finite field then the multiplicative part $F^\times$ is a cyclic group generated by $f$.
What - when nonzero - is $f^i + f^j$ as a power of $f$? What is 1,2,3,4,.. in terms of $f$?
For example $\mathbb F_{2^2} = \mathbb F_2[X]/(X^2+X+1)$ is generated by $X$ and $X^1 + X^2 = X^3$.
I don't think you can tell, in general. A finite field may have many generators, and it is possible that for generators $f$ and $g$ we have $1+f=f^i$ and $1+g=g^j$ with $i\ne j$. Perhaps you could try to find an example of this phenomenon.