Ramified primes of a mod-$\ell$ Galois representation (awaiting review)

If $\rho\colon\Gal_K\to G(\F_{\ell})$ is an mod-$\ell$ Galois representation with splitting field $L$, then a prime $\frak{p}\nmid\ell$ of $K$ is ramified if it is ramified in $L/K$.

Equivalently, a prime is ramified if the inertia subgroup of $\Gal_K$ for a prime above $\frak{p}$ is not contained in the kernel of $\rho$.

A prime which is not ramified is unramified.