Lower Bounds for the Least Prime in Chebotarev

TL;DR

This paper establishes a lower bound on the size of the smallest prime splitting completely in certain Galois number fields, demonstrating the optimality of existing upper bounds in Chebotarev density theorem.

Contribution

It proves the existence of infinite families of Galois number fields with minimal splitting primes matching known upper bounds, confirming their optimality.

Findings

Existence of infinite families with large minimal splitting primes

Lower bounds match known upper bounds up to small factors

Supports the sharpness of Chebotarev density bounds

Abstract

In this paper we show there exists an infinite family of number fields $L$, Galois over $\mathbb{Q}$, for which the smallest prime $p$ of $\mathbb{Q}$ which splits completely in $L$ has size at least $( \log(|D_L|) )^{2+o(1)}$. This gives a converse to various upper bounds, which shows that they are best possible.