I have a representation $\rho: G \rightarrow GL(V)$ where $G$ is a Galois group of an extension of $p$-adic fields ($p \neq 3$) and $V$ is a vector space over $\mathbb{Q}_3$.
Now I know that this representation is irreducible when we extend scalars to consider $V$ as a $\mathbb{C}$ vector space so does this mean it is irreducible over $\overline{\mathbb{Q}_3}$ as well?
I wish to use Schur's lemma to tell me that the image of some element is scalar but I am not sure if this scalar must live in $\mathbb{Q}_3$ since this field is not algebraically closed.