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.

Proof

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