  It is a classical problem in algebraic number theory to decide if a number
field is monogeneous, that is if it admits power integral bases. It is
especially interesting to consider this question in an infinite parametric
familiy of number fields. In this paper we consider the infinite parametric
family of simplest quartic fields $K$ generated by a root $\xi$ of the
polynomial $P_t(x)=x^4-tx^3-6x^2+tx+1$, assuming that $t>0$, $t\neq 3$ and
$t^2+16$ has no odd square factors. In addition to generators of power integral
bases we also calculate the minimal index and all elements of minimal index in
all fields in this family.
