Normal integral bases and Gaussian periods in the simplest cubic fields

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 that \(L_n/{\mathbb {Q}}\) is tamely ramified, which is a generalization of the work of Lehmer, Châtelet and Lazarus in the case that the conductor of \(L_n\) is equal to \(n^2+3n+9\).