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

Yoshichika Iizuka, Yutaka Konomi

TL;DR

This work examines whether classical invariants like the discriminant and regulator suffice to distinguish CM-fields. It constructs infinite families of imaginary biquadratic and imaginary cyclic quartic fields that share both discriminant and regulator, and in some cases also share class numbers, demonstrating limitations of zeta-function–based identifications. The authors develop and apply explicit polynomial models for cyclic quartic fields, analyze discriminants via known formulas, and show regulators can be forced to take arbitrarily large values by selecting suitable real quadratic subfields. Concrete examples illustrate the phenomena, underscoring that even equal regulators and discriminants do not force field isomorphism in higher degree cases.

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.

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

TL;DR

This work examines whether classical invariants like the discriminant and regulator suffice to distinguish CM-fields. It constructs infinite families of imaginary biquadratic and imaginary cyclic quartic fields that share both discriminant and regulator, and in some cases also share class numbers, demonstrating limitations of zeta-function–based identifications. The authors develop and apply explicit polynomial models for cyclic quartic fields, analyze discriminants via known formulas, and show regulators can be forced to take arbitrarily large values by selecting suitable real quadratic subfields. Concrete examples illustrate the phenomena, underscoring that even equal regulators and discriminants do not force field isomorphism in higher degree cases.

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.

Paper Structure

This paper contains 7 sections, 18 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Regulators of CM-fields
  3. Regulators of imaginary biquadratic fields
  4. Defining polynomials of cyclic quartic fields
  5. Discriminants of cyclic quartic fields
  6. The regulator of the real quadratic field $\mathbb{Q}(\sqrt{t^{2}+1})$
  7. Proofs of the main theorems and examples

Key Result

Theorem 1.1

The Dedekind zeta function $\zeta_{K}(s)$ of a number field $K$ converges absolutely for $\mathrm{Re}(s) > 1$ and extends to a meromorphic function on the entire complex plane $\mathbb{C}$, possessing only one simple pole at $s=1$ with residue where $r_{1}$ is the number of real embeddings and $2r_{2}$ is the number of complex embeddings of $K$.

Theorems & Definitions (40)

  • Theorem 1.1: Class Number Formula
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 2.1: Nakamula Nakamula1996, §2.4, Lemma 1
  • Theorem 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • ...and 30 more