The least non-split prime in a number field

Abstract

Let K be an $S_n$-field ($n\le 5$) with discriminant $d_K$. Let $n_K$ be the least prime which does not split completely in K. We prove unconditionally that $n_K=O(\log |d_K|)$, except for $O\left( X \text {exp}\left( -c\frac{\log X}{\log \log X}\right) \right)$ fields for some constant $c>0$. We also prove that the exceptional set is optimal, and that for any $\eta >0$, the $S_n$-fields such that $n_K\gg (\log |d_K|)^{1+\eta }$ are very rare.