The least unramified prime which does not split completely

Abstract

AbstractLet {K/F} be a finite extension of number fields of degree {n\geq 2}. We establish effective field-uniform unconditional upper bounds for the least norm of a prime ideal {\mathfrak{p}} of F which is degree 1 over {\mathbb{Q}} and does not ramify or split completely in K. We improve upon the previous best known general estimates due to Li [7] when {F=\mathbb{Q}}, and Murty and Patankar [9] when {K/F} is Galois. Our bounds are the first when {K/F} is not assumed to be Galois and {F\neq\mathbb{Q}}.