Let $K$ be the totally unramified extension of $\mathbf{Q}_p$ of degree $f$. Let $\sigma$ be the Frobenius of $K$.
Let $V$ be a $K$ vector space and $\varphi : V \to V$ be a $\sigma$-semi-linear bijection i.e. $\varphi$ is a bijective and additive map such that $\varphi(\lambda v) = \sigma(\lambda) v$ for all $\lambda\in K$ and $v\in V$. Then $\varphi^f$ is a linear automorphism of $V$ because $\sigma^f = \text{id}$. Let us suppose that $\varphi^f$ has all it's eigenvalues in $\mathbf{Q}_p$. Is it true that $V$ admits a full flag stable by $\varphi$?
Or, in other words, can we triangularize semi-linear automorphisms (under some conditions on eigenvalues)?