  Let $m$ be an integer, $m\neq -8,-3,0,5$ such that $m^2+3m+9$ is square free.
Let $\alpha$ be a root of \[ f=x^6-2mx^5-(5m+15)x^4-20x^3+5mx^2+(2m+6)x+1. \]
The totally real cyclic fields $K=Q(\alpha)$ are called simplest sextic fields
and are well known in the literature.
  Using a completely new approach we explicitly give an integral basis of $K$
in a parametric form and we show that the structure of this integral basis is
periodic in $m$ with period length 36. We prove that $K$ is not monogenic
except for a few values of $m$ in which cases we give all generators of power
integral bases.
