This question already has an answer here:
It is a standard result that finite fields have cyclic multiplicative groups (the nonzero elements with respect to field multiplication).
A recent discussion on this Question leads me to ask about the converse:
Is a field $F$ whose multiplicative group $F^*$ is cyclic necessarily finite?
Clearly the biggest a cyclic group can be is countable, and thus $F$ would also be at most countable.