Cyclic Cubic Field with Explicit Artin Symbols

Abstract

In this paper we present a set $\mathcal{T}_f^+$ of rational numbers $s\in\mathbf{Q}$ such that the minimal splitting fields $L_s$ of $X^3-3sX^2-(3s+3)X-1$ are cyclic cubic fields with a given conductor $f$. The set $\mathcal{T}_f^+$ has exactly one $s$ for each field $L$ of conductor $f$. The Weil's height of every number $s\in \mathcal{T}_f^+$ is minimal among all of the rational numbers $s\in\mathbf{Q}$ such that $L_s=L$. If a cyclic cubic field $L$ of conductor $f$ is given, then we can choose the number $s\in \mathcal{T}_f^+$ corresponding to $L$ by sequencing the explicit Artin symbols.