An infinite family of pairs of distinct quartic Galois CM-fields with the same discriminant and regulator

TL;DR

This paper constructs infinite families of pairs of distinct imaginary biquadratic and cyclic quartic fields sharing key invariants like discriminant, regulator, and class number, advancing understanding of their arithmetic properties.

Contribution

It introduces new infinite families of imaginary biquadratic and cyclic quartic fields with identical discriminant, regulator, and class number, highlighting their arithmetic similarities.

Findings

Constructed infinite families with same discriminant and regulator.

Provided examples with same discriminant, regulator, and class number.

Extended the understanding of invariants in imaginary Galois CM-fields.

Abstract

We construct an infinite family of pairs of distinct imaginary biquadratic fields and pairs of distinct imaginary cyclic quartic fields with the same discriminant and regulator. We also construct an infinite family of imaginary biquadratic fields and imaginary cyclic quartic fields with the same regulator. Moreover, we give examples of a pair of distinct imaginary biquadratic fields and a pair of distinct imaginary cyclic quartic fields with the same discriminant, regulator and class number.