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