I have the following problem:
Let $V$ be a one dimensional vector space over $k = \mathbb{F}_{p^e}$. Show that there are precisely $p^e$ distinct non-isomorphic $k[x]$-module structures on $V$.
This is what I have so far. Let $v$ be a basis for $V$. We know that to give $V$ a $k[x]$-module structure, we need to find a linear transformation $T:V \rightarrow V$ and see how $T$ acts on $V$. So, I will find the distinct module structures if I see how any such linear transformation acts on the generator $v$. Now, I am having troubles finding such a linear transformation.
First of all, is this a good approach? If not, what other approach could I use? Does anyone have any hint or could help me out? Thanks so much for your help!