# elementary abelian groups and finite fields

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

• Yes, the underlying abelian group of any finite field is elementary abelian. – Tobias Kildetoft Feb 6 '14 at 10:41

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 $\;pa=0\;\;\forall\,a\in GF(p^n)\;$ ...
• Those are not the elements of $\;GF(4)\;$ , @user118746 ...!I'm afraid you're confussing the field with 4 elements and the non-field ring $\;\Bbb Z/4\Bbb Z=:\Bbb Z_4\;$ with four elements. – DonAntonio Feb 6 '14 at 12:08
• Yes to both your questions. Only when $\;n=1\;$ we have that equality $\;GF(p)=\Bbb Z/p\Bbb Z\;$, @user118746 – DonAntonio Feb 6 '14 at 12:28
• @user118746: Look up this question for the arithmetic of $GF(4)$. Some aspects of the arithmetic operations in $GF(8)$ and $GF(16)$ are spelled out here. – Jyrki Lahtonen Feb 6 '14 at 12:33