Let $F$ be a field with prime $p$ characteristic and let $X$ be a periodic subgroup of $(F,\, \cdot)$. Let now $K$ be the subfield of $F$ generated by the elements of $X$.
Is it possible to describe $K$ as the set of linear combinations of elements of $X$ by coefficients in $GF(p)$?
In particular is it possible to express an inverse of these linear combinations as another linear combination?
If it is true, can someone give me some clue?
I think the answer is yes if $X$ has bounded period, for then $X$ is necessarily finite and cyclic. If $X$ has order $d > 1$, choose $\alpha \in F$ of order $d,$ and let $\alpha$ have minimum polynomial $f(x)$ over $K = {\rm GF}(p).$ Then $f(x)$ divides $x^{d}-1$, and the degree of $f(x)$ is $m,$ the smallest positive integer such that $d$ divides $p^{m}-1$. Now every positive power of $\alpha$ is a $K$-linear combination of $\{1,\alpha,\ldots, \alpha^{m-1} \}$ and any product of $K$-combinations of elements of $X$ is still a $K$-linear combination of $\{1,\alpha,\ldots, \alpha^{m-1} \}.$ In particular, every element of $X$ is such a combination. Then it is clear that the subfield of $F$ generated by $X$ is contained in $K[\alpha]$, and hence is equal to that field.