Normal integral bases and Gaussian periods in the simplest cubic fields.

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