Equivalence of definitions of additive group of a field

Statement
An Abelian group that is FC-simple, i.e., it has no proper nontrivial fully characteristic subgroup, occurs as the additive group of a field.

Facts used

 * For any integer $$n$$, the set of elements of the form $$ng$$, i.e., the set of $$n^{th}$$ powers in the group, is a fully characteristic subgroup.
 * For any integer $$n$$, the set of elements $$g$$ for which $$ng = 0$$, i.e., the set of $$n^{th}$$ roots of unity in the group, is a fully characteristic subgroup.