Lower Bounds for the Least Prime in Chebotarev

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