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

TL;DR

This paper investigates the minimal index and all elements of minimal index in an infinite family of simplest quartic fields, providing explicit calculations and characterizations for these number fields.

Contribution

It introduces explicit formulas and methods for calculating the minimal index and elements of minimal index in an infinite parametric family of simplest quartic fields.

Findings

Explicit formulas for minimal index in the family.

Complete characterization of elements of minimal index.

Conditions under which the fields are monogeneous.

Abstract

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.