Determinant of an Artin representation (reviewed)

Given an Artin representation $\rho:\Gal(\overline{\Q}/\Q)\to\GL(V)$, where $V$ is a finite dimensional vector space over $\C$, we can compose with the $\det:\GL(V)\to \C^\times$ to get a one-dimensional Artin representation. This one-dimensional representation is called the determinant of $\rho$.