As the title states, I'm having trouble considering a finite field $\mathbb{F}_{p^n}$ as a vector space V over $\mathbb{F}_p$. Clearly it is dimension $n$. How can we work with this vector space? Specifically, I am trying to find the the characteristic polynomial of $\sigma_p : x \rightarrow x^p$ and deduce some things about it, but I'm not sure how to grasp it.
So I don't even know how to think about the bases. If $\mathbb{F}_p = \{a_1, a_2, ... a_p\},$ then is $\mathbb{F}_{p^n} = \{a_1, a_1^2, ... a_1^n, a_2, a_2^2, .... a_p^n\}$?
Anything helps, thanks.
CK
The elements of $\mathbb{F}_{p^n}$ are the roots of $$f(x) = x^{p^n} -x \in \mathbb{F}_{p}[x]$$ To understand this recall that the multiplicative group of $\mathbb{F}_{p^n}$ is cyclic of order $p^n -1$.
This implies that $$\sigma_{p}^{n} = 1$$ and so the characteristic polynomial of $\sigma_p$ is $x^n -1 $
For the second point: $\mathbb{F}_{p^n}$ is a vector space over $\mathbb{F}_{p}$ of dimension $n$ so if $\lbrace v_1 \ldots v_n \rbrace$ is a basis $$\mathbb{F}_{p^n} = \lbrace \sum_{j = 1}^{n} r_{j}v_j | \ \ r_j \in \mathbb{F_p}\rbrace$$
The set you wrote is not correct, for example because it contains $np$ elements and not $p^n$