  Let $n>1$, $e\geq 0$ and a prime number $p\geq 2^{n+2+2e}+3$, such that the
index of regularity of $p$ is $\leq e$. We show that there are infinitely many
irreducible Galois representations $\rho:
Gal(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow {GL}_n(\mathbb{Q}_p)$ unramified at
all primes $l\neq p$. Furthermore, these representations are shown to have
image containing a fixed finite index subgroup of ${SL}_n(\mathbb{Z}_p)$. Such
representations are constructed by lifting suitable residual representations
$\bar{\rho}$ with image in the diagonal torus in ${GL}_n(\mathbb{F}_p)$, for
which the global deformation problem is unobstructed.
