An explicit upper bound for the least prime ideal in the Chebotarev density theorem

Abstract

Lagarias, Montgomery, and Odlyzko proved that there exists an effectively computable absolute constant $A_1$ such that for every finite extension $K$ of ${\mathbb{Q}}$, every finite Galois extension $L$ of $K$ with Galois group $G$ and every conjugacy class $C$ of $G$, there exists a prime ideal $\mathfrak{p}$ of $K$ which is unramified in $L$, for which $\left[\frac{L/K}{\mathfrak{p}}\right]=C$, for which $N_{K/{\mathbb Q}}\,\mathfrak{p}$ is a rational prime, and which satisfies $N_{K/{\mathbb Q}}\,{\mathfrak{p}} \leq 2 {d_L}^{A_1}$. In this paper we show without any restriction that $N_{K/{\mathbb Q}}\,{\mathfrak{p}} \leq {d_L}^{12577}$ if $L \neq {\mathbb Q}$, using the approach developed by Lagarias, Montgomery, and Odlyzko.