Let $F$ be a nonarchimedean local field. Does there exist a semisimple algebraic group over $F$ which is not split over a maximal unramified extension of $F$ ?



Semisimple group not split by an unramified extension?
The Galois group of $F$ maps onto $S\_3$, the symmetric group on three letters. The group $S\_3$ is also the group of diagram automorphisms of the Dynkin diagram of $SO(8)$. That means that there is a quasi-split group $G$ defined over $F$, which over an extension $E/F$ with Galois group $S\_3$, becomes the split group $SO(8)$ over $E$. The group $G$ cannot split over an unramified extension $L/F$, because its Galois group would be abelian, and cannot cover all the diagram automorphisms. 


