Example of infinite field $(F,+,.)$ such that $(F^*, . )$ is a cyclic group ?

Question

It is known that any infinite cyclic group can never be a vector space , from this we can derive that if $(F,+,.)$ is an infinite field then $(F,+)$ cannot be cyclic . I am asking , is there any infinite field $(F,+,.)$ such that $(F^*, . )$ is a cyclic group ? (, where $F^*:=F$\ $\{0_F\}$ )

Answer 1

Such an $F$ could not be characteristic zero, as then it would contain $\Bbb Q$, and $\Bbb Q^\times$ isn't cyclic - and the only subgroups of cyclic groups (e.g. $F^\times$) are cyclic. So $F$ would be characteristic $p>0$; it cannot be an algebraic extension of $\Bbb F_p$, as any element $x$ algebraic over $\Bbb F_p$ has finite order (because $\Bbb F_p(x)$ is finite dimensional over $\Bbb F_p$, hence finite). So $F$ would necessarily have an element transcendental over $\Bbb F_p$ and thus $\Bbb F_p(X)$ is isomorphic to a subfield of $F$. Now you can check that $\Bbb F_p(X)^\times$ is not cyclic: consider for instance the subgroup generated by $X$ and $X+1$ and show that it's not cyclic.