Bounding the least prime ideal in the Chebotarev Density Theorem

TL;DR

This paper provides explicit bounds on the smallest prime ideal in Galois extensions of number fields using Chebotarev's theorem, and explores a quantitative Deuring-Heilbronn phenomenon for Dedekind zeta functions.

Contribution

It unconditionally bounds the least prime ideal in Galois extensions as a power of the discriminant with an explicit exponent and establishes a quantitative Deuring-Heilbronn phenomenon.

Findings

Bound on the least prime ideal as a power of the discriminant

Explicit exponent in the bound

Quantitative Deuring-Heilbronn phenomenon for Dedekind zeta function

Abstract

Let $L$ be a finite Galois extension of the number field $K$. We unconditionally bound the least prime ideal of $K$ occurring in the Chebotarev Density Theorem as a power of the discriminant of $L$ with an explicit exponent. We also establish a quantitative Deuring-Heilbronn phenomenon for the Dedekind zeta function.