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Let $F$ be a locally compact non-archimedean field of residue characteristic$p$, $\textbf{G}$ a connected reductive group over $F$, and $R$ a field ofcharacteristic $p$. When $R$ is algebraically closed, the irreducibleadmissible $R$-representations of $G=\textbf{G}(F)$ are classified in term ofsupersingular $R$-representations of the Levi subgroups of $G$ and parabolicinduction; there is a similar classification for the simple modules of thepro-$p$ Iwahori Hecke $R$-algebra. In this paper, we show that bothclassifications hold true when $R$ is not algebraically closed.

Representations of a $p$-adic group in characteristic $p$