Galois orbit of an Artin representation (reviewed)

The absolute Galois group $\Gal(\overline\Q/\Q)$ acts on the set of Artin representations for an extension by pre-composition, so we consider two representations to be Galois conjugate if they differ by such an automorphism. The action, for a given representation, in fact factors through a finite degree abelian extension.

The Galois orbit of a representation is its orbit under this action.