In general, we know that if $G$ is a finite group and $K$ is a field, then $K[G]$ (the group algebra) is semisimple whenever $\operatorname{char}(K)$ does not divide the order of $G$. However, this result does not hold when we have a $p$-group and a finite field of characteristic $p$.
How would I go about showing that an irreducible representation of a $p$-group $G$ over a field $K$ of characteristic $p$ must be the trivial representation?
It seems like this would perhaps entail some sort of application of Maschke's Theorem to get a contradiction. I know that $|G| = \sum_i \operatorname{dim}(V_i)^2$, where $V_i$ is an irreducible representation. But it seems like that does not help me much here.
Any thoughts as to how to approach this question?