Normal integral bases and Gaussian periods in the simplest cubic fields

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 $n^2+3n+9$.