The smallest prime that does not split completely in a number field

TL;DR

This paper investigates bounds on the smallest prime that does not split completely in a number field, extending classical non-residue problems, using Dedekind zeta functions and multiplicative functions, providing improved bounds for various fields.

Contribution

The paper introduces two new approaches to bounding the least non-split prime in a number field, improving existing bounds across different degrees.

Findings

Derived the best known bounds for all number fields with degree > 2.

Established new upper bounds for the Dedekind zeta function residue for small degree fields.

Connected the problem to properties of multiplicative functions, offering novel insights.

Abstract

We study the problem of bounding the least prime that does not split completely in a number field. This is a generalization of the classic problem of bounding the least quadratic non-residue. Here, we present two distinct approaches to this problem. The first is by studying the behavior of the Dedekind zeta function of the number field near 1, and the second by relating the problem to questions involving multiplicative functions. We derive the best known bounds for this problem for all number fields with degree greater than 2. We also derive the best known upper bound for the residue of the Dedekind zeta function in the case where the degree is small compared to the discriminant.