Let $F$ be a field of order $p^n$ for some prime $p$ and positive integer $n$, and let $\mathbb{Z}_{p} \subset F$ be a prime field of $F$.
Prove the additive group of $F$, that is the group $(F,+)$ is isomorphic to $(\mathbb{Z}_{p}, +) \times (\mathbb{Z}_{p}, +) \times ... \times (\mathbb{Z}_{p},+)$ ($n$ times)
I'm clueless. I see $(\mathbb{Z}_{p}, +) \times (\mathbb{Z}_{p}, +) \times ... \times (\mathbb{Z}_{p},+)$ ($n$ times) has order $p^n$, but I can't see why $F$ would. Maybe it is because of the fact that every element of $F$ is a root of $x^{p^n} -x$ which has degree $p^n$ and therefore $p^n$ roots. But then what is the operation of $(\mathbb{Z}_{p}, +) \times (\mathbb{Z}_{p}, +) \times ... \times (\mathbb{Z}_{p},+)$ ($n$ times) which makes it a group?
It then seems obvious that every root of $F$ can be mapped to a $n$-dimensional vector modulus $p$, which I guess $(\mathbb{Z}_{p}, +) \times (\mathbb{Z}_{p}, +) \times ... \times (\mathbb{Z}_{p},+)$ ($n$ times) is reffering to, but I can't see how to do this systematically, i.e create a bijection, let alone test the homomorphism properties (I've forgotten them exactly), as an unsure of the operation of $(\mathbb{Z}_{p}, +) \times (\mathbb{Z}_{p}, +) \times ... \times (\mathbb{Z}_{p},+)$ ($n$ times)
Any pointers?
Since $F$ has characteristic $p$, $(F,+)$ is an abelian group with the property that $px = 0$ for all $x \in F$. So $F$ has the structure of a vector space over $\Bbb Z_p$. It follows that $(F,+)$ is isomorphic (as a $\Bbb Z_p$-vector space) to $(\Bbb Z_p^m,+)$ for some $m$. So $|F| = |\Bbb Z_p^m| = p^m$, whence $m = n$. In particular, $(F,+)$ is isomorphic (as an abelian group) to $(\Bbb Z_p, +) \times \cdots \times (\Bbb Z_p, +)$ ($n$ times).