201011.4492v1 On the second smallest prime non-residue

Let χ be a non-principal Dirichlet character modulo a prime p. Let q1 < q2 denote the two smallest prime non-residues of . We give explicit upper bounds on q2 that improve upon all known results. We also provide a good upper estimate on the product q1q2 which has an upcoming application to the study of norm-Euclidean Galois fields.Keywords: Dirichlet chara..