Image of the representation (reviewed)

An Artin representation $\rho:\textrm{Gal}(\overline{\Q}/\Q)\to \GL(V)$ factors through the Galois group of a finite extension of $\Q$. Thus its image is a finite group $G \subset \GL(V)$ which is canonically isomorphic to the Galois group of a finite extension $K/\Q$ where $K$ is the fixed field of $\ker(\rho)$ and is called the Artin field of $\rho$. We sometimes say that $\Gal(K/\Q)$ is the image of the Artin representation $\rho$.