Linnik [15] proved that each arithmetic progression modulo m contains a prime number p < 2m c , where c is an absolute constant. We consider here t he problem of generalizing Linnik's theorem to an algebraic number field K in an arithmetically invariant way i.e. making the dependence on K explicit. If m is an integral ideal of K then an "arithmetic progression" modulo m is a coset of the group P m of ray classes mod m in the group /(m) of fractional ideals of K which are coprime to m. In these terms we will prove Theorem. Lei K be a number field of absolute degree n over 0 and absolute discriminant d K . Denote the absolute norm by N. lfm is an integral ideal of K then each coset ofP m in /(m) contains a prime ideal p ofK withwhere a, b are effectively computable positive constants independent ofK. This problem was considered by Fogels [5], [6], [7] who obtained the theorem with "constants" depending on n in an unspecified manner. Actually our results (5. 2), (5. 3) contain rat her more than the above theorem in considering also the existence of prime ideals in "short intervals" and also a lower bound for their number, all in a A^-uniform formulation. Also the results are formulated in terms of better behaved invariants than above, in a sense to be discussed below; the above theorem can be recovered from (5. 3) by using (1. 16).Specializing the theorem to K=Q yields Linnik's theorem. In a quite different direction specializing t o m = (1) gives Corollary. Each ideal class ofK contains a prime ideal p withIn favorable cases this corollary, like Linnik's theorem, fails to describe the actual phenomenon only because the constants are too large: see (5.4).There is an even broader generalization of Linnik's problem: if L/F is a finite Galois extension of number fields then the primes of F are partitioned into "Tchebotarev classes" in accordance with the partition of gal(L/F) into conjugacy classes, and we are to find an upper bound, depending on the arithmetic of L/F, for the "least prime"

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Abstract.Let G be a finite group. An effective criterion is given for a ZGlattice to be a direct summand of a permutation lattice.

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