Explicit bounds for primes in residue classes

Let E/K be an abelian extension of number fields, with E 6= Q. Let ∆ and n denote the absolute discriminant and degree of E. Let σ 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 ) = σ, satisfying Np ≤ (1 + o(1))(log ∆ + 2n)2. (2) There is a degree-1 prime p of K such that ( p E/K ) generates the same group as σ, satisfying Np ≤ (1 + o(1))(log ∆)2. (3) For K = Q, there is a prime p such that ( p E/Q ) = σ, satisfying p ≤ (1 + o(1))(log ∆)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 ≤ (1 + o(1))(φ(q) log q)2. It follows from our proof that (1)–(3) also hold for arbitrary Galois extensions, provided we replace σ by its conjugacy class 〈σ〉. 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.