  Let $K$ be a fixed number field, assumed to be Galois over $\mathbb Q$. Let
$r$ and $f$ be fixed integers with $f$ positive. Given an elliptic curve $E$,
defined over $K$, we consider the problem of counting the number of degree $f$
prime ideals of $K$ with trace of Frobenius equal to $r$. Except in the case
$f=2$, we show that "on average," the number of such prime ideals with norm
less than or equal to $x$ satisfies an asymptotic identity that is in
accordance with standard heuristics. This work is related to the classical
Lang-Trotter conjecture and extends the work of several authors.
