Normal integral bases and Gaussian periods in the simplest cubic fields

TL;DR

This paper classifies all normal integral bases for the simplest cubic fields generated by Shanks' polynomial when they are tamely ramified, and relates roots to Gaussian periods, extending previous work.

Contribution

It provides a complete description of normal integral bases for these cubic fields and establishes explicit relations with Gaussian periods in the tamely ramified case.

Findings

All normal integral bases for the simplest cubic fields are characterized.

Explicit relations between roots and Gaussian periods are derived.

The results generalize previous work by Lehmer, Châtelet, and Lazarus.

Abstract

We give all normal integral bases for the simplest cubic field $L_n$ generated by the roots of Shanks' cubic polynomial when these bases exist, that is, $L_n/\mathbb Q$ is tamely ramified. Furthermore, as an application of the result, we give an explicit relation between the roots of Shanks' cubic polynomial and the Gaussian periods of $L_n$ in the case $L_n/\mathbb Q$ is tamely ramified, which is a generalization of the work of Lehmer, Ch\^{a}telet and Lazarus in the case that the conductor of $L_n$ is equal to $n^2+3n+9$.