Let $ L $ be a finite extension of $p$-adic numbers $ \mathbb{Q}_p $. Let $ \text{GL}_{n}(L) $ denote the general linear group $ \text{GL}_{n}(L) $ over $L$ equipped with the topology induced from the $p$-adic field $ L$. An element $ g\in \text{GL}_{n}(L) $ is called *regular semisimple* if it has pairwise distinct eigenvalues in the algebraic closure $ \overline{\mathbb{Q}}_{p} $ of $\mathbb{Q}_p$. A *linear group* over $L$ is a subgroup of $ \text{GL}_{n}(L) $.

My question is the following: When does a linear group $ \Gamma $ have a regular semisimple element?

As pointed out in the comments, it is not the case only under the assumption that $ \Gamma $ has no open solvable subgroup.

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