# Additive subgroup of a field of characteristic $p$ is an elementary abelian $p$-group

In this paper (in the abstract), it is mentioned that:

A finite subgroup of the additive group of a field $F$ of characteristic $p \neq 0$ is an elementary abelian $p$-group.

Why is this so?

If $charF=p$, then $p=0$ in $F$ so $px=0$ for every $x\in F$.
An elementary abelian p-group is an abelian group in which every nontrivial element $x$ has order $p$ (namely $px=0$) by definition.