Explicit counting of ideals and a Brun-Titchmarsh inequality for the Chebotarev density theorem

We prove a bound on the number of primes with a given splitting behavior in a given field extension. This bound generalizes the Brun-Titchmarsh bound on the number of primes in an arithmetic progression. The proof is set up as an application of Selberg's Sieve in number fields. The main new ingredient is an explicit counting result estimating the number of integral elements with certain properties up to multiplication by units. As a consequence of this result, we deduce an explicit estimate for the number of ideals of norm up to x.