Characteristic polynomial of Frobenius (awaiting review)

Let $\rho:\Gal_K\to G(\F_{\ell})$ be a mod-$\ell$ Galois representation. The characteristic polynomial of Frobenius of a prime $\frak{p}$ of $K$ at which $\rho$ is unramified is the characteristic polynomial of $\rho(\text{Frob}_{\frak{p}})$.

Note that unramified primes $\frak{p}$ of $K$ are, by definition, unramified in the splitting field $L/K$ of $\rho$, so the Frobenius automorphism $\text{Frob}_{\frak{p}}$ is well-defined up to conjugacy in $\Gal(L/K)$, and hence the characteristic polynomial of $\rho(\text{Frob}_{\frak{p}})$ is well-defined.