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/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/Q is tamely ramified, which is a generalization of the work of Lehmer, Chatelet and Lazarus in the case that the conductor of L-n is equal to n(2) + 3n + 9.