Let $p$ be a prime. My eventual goal is to prove that the only irreducible representation of a $p$-group over a field of characteristic $p$ is the trivial representation.
At the moment, I'm trying to prove a simpler claim: suppose $G$ is a (cyclic) group of order $p$, let $K$ be a field of characteristic $p$, and let $V$ be any representation of $G$ over $K$; if the dimension of $V$ (as a vector space over $K$) is greater than $1$, then $V$ must be a reducible representation.
Does anyone have any hints or suggestions on how to proceed? For example, where should I use the fact that $K$ has characteristic equal to $|G|$ (actually, I'm not even sure whether or not the above holds for arbitrary $K$) or the fact that $G$ is cyclic?