  For a fixed number field $K$, we consider the mean square error in estimating
the number of primes with norm congruent to $a$ modulo $q$ by the Chebotar\"ev
Density Theorem when averaging over all $q\le Q$ and all appropriate $a$. Using
a large sieve inequality, we obtain an upper bound similar to the
Barban-Davenport-Halberstam Theorem.
