Representations of a cyclic group of order p over a field of characteristic p?
Here are some hints:
- How many elements does $V$ have? (I am assuming without loss of generality that it is finite-dimensional.)
- What do you know about the sizes of orbits of the vectors in $V$ under the action of $G$?
If you can answer these questions, you will immediately be able to prove that the only irreducible representation is the trivial one.
In your special case, you can show that the group algebra of a cyclic group of order $p$ over a field $k$ of characteristic $p$ is isomorphic to $k[t]/(t^p)$. It follows that understanding $G$-modules is the same as understanding nilpotent endomorphisms of vector spaces of nilpotency index at most $p$: this can be done using Jordan canonical forms. Indeed, the Jordan form of the matrix of $t$ in a simple module has only one block, because the module is indescomposable, has only zero as eigenvalue because it is nilpotent, and you should have little trouble showing that that block must be of size $1$.