The least unramified prime which does not split completely

Abstract

Let $K/F$ be a finite extension of number fields of degree $n\ge 2$. We establish effective field-uniform unconditional upper bounds for the least norm of a prime ideal 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 X. Li when $F=\mathbb{Q}$ and Murty-Patankar when $K/F$ is Galois. Our bounds are the first when $K/F$ is not assumed to be Galois and $F\ne \mathbb{Q}$.

13 pages

References in corpus (1)

• An explicit bound for the least prime ideal in the Chebotarev density theorem