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

Explicit counting of ideals and a Brun-Titchmarsh inequality for the Chebotarev Density Theorem