elementary abelian groups and finite fields

Question

in group theory, an elementary abelian group is a finite abelian group, where every nontrivial element has order $p$, where $p$ is a prime; it is a particular kind of $p$-group. now suppose that we have a finite field $GF(p^n)$ and we want to consider it as a vector space over $GF(p)$. Can we claim that $GF(p^n)$ is an elementary abelian group? I have already read the answer of "Amitesh Datta". Finite fields as vector spaces

Answer 1

An elementary abelian group $\;G\;$ is one for which $\;pg=0\;$ for some prime $\;p\;$ and for all $\;g\in G\;$, i.e. any non-trivial element has (additive, in my writing) order equal to $\;p\;$

Now ask yourself whether it is true that $\;pa=0\;\;\forall\,a\in GF(p^n)\;$ ...