Multiplicative group of a field implies every finite subgroup is cyclic

Statement
The multiplicative group of a field is a group in which every finite subgroup is cyclic: in other words, every finite subgroup of the multiplicative group of a field is a cyclic subgroup.

(Note that the result holds more generally for the multiplicative group of an integral domain).

For finite fields

 * Multiplicative group of a prime field is cyclic
 * Multiplicative group of a finite field is cyclic

For infinite fields

 * Classification of fields whose multiplicative group is locally cyclic
 * Classification of fields whose multiplicative group is uniquely divisible

For commutative rings that are not fields

 * Classification of natural numbers for which the multiplicative group is cyclic

Noncommutative analogues

 * Multiplicative group of a skew field implies every finite abelian subgroup is cyclic

Facts used

 * 1) uses::Multiplicative group of a field implies at most n elements of order dividing n
 * 2) uses::At most n elements of order dividing n implies every finite subgroup is cyclic

Proof
The proof follows directly by piecing together facts (1) and (2).