# The least unramified prime which does not split completely

## TL;DR

This paper provides effective, unconditional upper bounds for the smallest prime ideal in a number field that does not ramify or split completely in an extension, improving previous estimates especially when the extension is not Galois.

## Contribution

It introduces the first bounds for such primes in non-Galois extensions over arbitrary number fields, extending prior work limited to Galois cases or rational fields.

## Key findings

- First unconditional bounds for non-Galois extensions
- Improved estimates over previous results for specific cases
- Applicable to general number field extensions

## Abstract

Let $K/F$ 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 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 $F \neq \mathbb{Q}$.