Let $p$ be a prime number, $\mathbb{Q}\_p$ the field of $p$-adic numbers, $G\_p$ the absolute Galois group of $\mathbb{Q}\_p$ and $V$ a finite dimensional vector space over $\mathbb{Q}\_p$. Assume we are given a linear representation $\rho : G\_p \to GL(V)$.


Can we find a closed subgroup of finite index $H$ in $G\_{p}$ such that the restriction of $\rho$ to $H$ is reducible ?


It is possible in the case of a representation which have coefficients in $\mathbb{F}\_p$ (in this case, absolutely irreducible representations are induced).



Absolutely irreducible p-adic representation of the absolute Galois group of Q_p
The answer is no. To prove it, it suffices to show that there exists a representation $\rho: G\_p \rightarrow Gl(V)$ of open image. There are several ways to do this, one is to use Chenevier prop. 1.8 in "Quelques courbes de Hecke se plongent dans l'espace de Colmez", which says that for $p=2,3,5,7$ and
$\rho: G\_{\mathbb Q,p} \rightarrow Gl\_2(\mathbb Q\_p)$ an odd representation (where $G\_{\mathbb Q,p}$ is the global galois group of $\mathbb Q$ unramified outside $p$),
one has $\rho(G\_{\mathbb Q,p})=\rho(G\_p)$. So it suffices to find now a "global" representation with open image, and this is well-known to exist (e.g. Serre's open image for elliptic curve, etc.)


