  We prove a prime number theorem first for the classical Rankin-Selberg
L-function $L(s,\pi\times\pi')$ over any Galois extension with $\pi$ and $\pi'$
unitary automorphic cuspidal representations of $GL_n$ and $GL_m$ respectively
with at least one of the representations subject to a self-contragredient
assumption. We then extend these results to two representations $\pi$ defined
on $GL_n/E$ and $\pi'$ defined on $GL_m/F$ with $E$ and $F$ cylic algebraic
number fields of coprime degree where $\pi$ and $\pi'$ admit a base change lift
from $\mathbb{Q}$ again given a self-contragredient assumption on at least one
of the representations which lift to $\pi$ or $\pi'$.
