Lower bounds for the least prime in Chebotarev

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 ${\left(log\left(\left|D_L\right|\right)\right)}^{2+o\left(1\right)}$. This gives a converse to various upper bounds, which shows that they are best possible.