The multiplicative group of $\mathbb{F}_{p^n}$ is the circle group $C _{p^n-1}$. So, each nonzero element corresponds to a rotation, and multiplication composes rotations.
The field is also an extension field of $\mathbb{F}_{p}$, which is equivalent to the field of integers mod $p$ with normal addition and multiplication. Here, the additive group is $C _{p}$ and the multiplicative group is $C _{p-1}$.
The field extension is a vector space over $\mathbb{F}_{p}$, so each addition is the same as pointwise addition over $C _{p}^n$, and explains why repeated addition will always take equal elements to $0$, however the field structure somehow "joins" these together into one larger circle group under multiplication.
How does the field structure link $C _{p}^n$ and $C _{p^n}$? Clearly $(0, 0, ..., 0)$ are joined together as the field additive identity, and multiplication is an interaction between these groups, but I can't see much more than that. I'm especially interested in fields of order $2^n$, even more so $2^{2^n}$. I wouldn't be surprised if different orders lead to very different structures.