On Galois representations with large image

For every prime number $pgeq 3$ and every integer $mgeq 1$, we prove the existence of a continuous Galois representation $rho: G_mathbb{Q} rightarrow Gl_m(mathbb{Z}_p)$ which has open image and is unramified outside ${p,infty}$ (resp. outside ${2,p,infty}$) when $pequiv 3$ mod $4$ (resp. $p equiv 1$ mod $4$).