Let E/K be an abelian extension of number fields, with E ¬= Q. Let Δ 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))(logΔ + 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 Δ) 2 . (3) For K = Q, there is a prime p such that (p/E/Q) = σ, satisfying P ≤ (1 + o(1))(log Δ) 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 . 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 .