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

Calculating all elements of minimal index in the infinite parametric family of simplest quartic fields