A family of cyclic quartic fields with explicit fundamental units

TL;DR

This paper constructs a family of cyclic quartic fields and explicitly identifies their fundamental units or subgroups of units, advancing understanding of unit groups in these number fields.

Contribution

It provides explicit constructions of cyclic quartic fields with detailed descriptions of their fundamental units or unit subgroups.

Findings

Constructed a family of cyclic quartic fields.

Identified roots as fundamental units or subgroup generators.

Analyzed the index of the subgroup generated by roots.

Abstract

We construct a family of quartic polynomials with cyclic Galois group and show that the roots of the polynomials are fundamental units or generate a subgroup of index 5.