Explicit bounds for primes in residue classes

Let E/K be an abelian extension of number fields, with E not equal Q. Let Delta and n denote the absolute discriminant and degree of E. Let sigma denote an element of the Galois group of E/K. We prove the following theorems, assuming the Extended Riemann Hypothesis: (1) There is a degree-1 prime p of K such that (p/E/K) = sigma, satisfying N-p less than or equal to (1 + o(1))(log Delta + 2n)(2). (2) There is a degree-1 prime p of K such that (p/E/K) generates the same group as sigma, satisfying N-p less than or equal to (1 + o(1))(log Delta)(2). (3) Far K = Q, there is a prime p such that (p/E/Q) = sigma, satisfying P less than or equal to (1 + o(1))(log Delta)(2). In (1) and (2) we can in fact take p to be unramified in K/Q. A special case of this result is the following. (4) If gcd(m, q) = 1, the least prime p = m (mod q) satisfies P less than or equal to (1 + o(1))(phi(q) log q)(2). It follows from our proof that (1)-(3) also hold for arbitrary Galois extensions, provided we replace sigma by its conjugacy class [sigma].. Our theorems lead to explicit versions of (1)-(4), including the following: the least prime p = m (mod q) is less than 2(q log q)(2).