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The unit group and the 2-class number of some fields of the form $\mathbb{Q}(\sqrt{2}, \sqrt{pq}, \sqrt{ps})$ and $\mathbb{Q}(\sqrt{2}, \sqrt{pq}, \sqrt{ps}, \sqrt{-\ell})$

Moha Ben Taleb El Hamam

TL;DR

This work addresses the explicit computation of the unit group and the $2$-class number for multiquadratic fields of the form $L^+=\mathbb{Q}(\sqrt{2},\sqrt{pq},\sqrt{ps})$ and $L=\mathbb{Q}(\sqrt{2},\sqrt{pq},\sqrt{ps},\sqrt{-\ell})$, under specific congruence and Legendre-symbol conditions on three distinct primes $p,q,s$ and odd square-free $\ell$. The authors deploy Wada's method for multiquadratic fields, together with detailed norm calculations from biquadratic subextensions, to construct explicit fundamental units and to determine the $2$-class group structure. They present complete unit descriptions for both cases (case1 and case2) of the prime-symbol conditions, distinguishing the role of $\ell$ (whether $\ell=1$ or not) in the structure of $E_{L}$ and the presence of $\zeta_8$-factors. Additionally, they derive the $2$-class-number implications (e.g., $\mathrm{Cl}_2(\mathbb{L}^+) \cong (\mathbb{Z}/2\mathbb{Z})^2$ in one scenario) and discuss how these results inform the study of $2$-class field towers and cyclotomic $\mathbb{Z}_2$-extensions related to the base biquadratic fields.

Abstract

Let $\LL^+=\mathbb{Q}(\sqrt{2}, \sqrt{pq}, \sqrt{ps})$ and $\LL=\mathbb{Q}(\sqrt{2}, \sqrt{pq}, \sqrt{ps}, \sqrt{-\ell})$ be two fields, where $q$, $p$ and $s$ three different prime integers and $\ell\geq1$ be a positive odd square-free integer relatively prime to $q$, $p$ and $s$. The purpose of this paper is to show how one can proceed to perform the calculation of unit group of the fields of the form $\LL^+=\mathbb{Q}(\sqrt{2}, \sqrt{pq}, \sqrt{ps})$ and $\LL=\mathbb{Q}(\sqrt{2}, \sqrt{pq}, \sqrt{ps}, \sqrt{-\ell})$. More precisely, we compute the unit group and the $2$-class number of these fields whenever $p\equiv-s\equiv 5\pmod 8, q\equiv7\pmod 8 ~~ \text{and} ~~ \left(\frac{ p}{ q}\right)=\left(\frac{ p}{ s}\right)=\left(\frac{s}{ q}\right)=1$ and $\left(\frac{ p}{ q}\right)=\left(\frac{ p}{ s}\right),$ or $p\equiv-s\equiv 5\pmod 8, q\equiv7\pmod 8 ~~ \text{and} ~~ \left(\frac{ p}{ q}\right)=\left(\frac{ p}{ s}\right)=\left(\frac{s}{ q}\right)=-1$.

The unit group and the 2-class number of some fields of the form $\mathbb{Q}(\sqrt{2}, \sqrt{pq}, \sqrt{ps})$ and $\mathbb{Q}(\sqrt{2}, \sqrt{pq}, \sqrt{ps}, \sqrt{-\ell})$

TL;DR

This work addresses the explicit computation of the unit group and the -class number for multiquadratic fields of the form and , under specific congruence and Legendre-symbol conditions on three distinct primes and odd square-free . The authors deploy Wada's method for multiquadratic fields, together with detailed norm calculations from biquadratic subextensions, to construct explicit fundamental units and to determine the -class group structure. They present complete unit descriptions for both cases (case1 and case2) of the prime-symbol conditions, distinguishing the role of (whether or not) in the structure of and the presence of -factors. Additionally, they derive the -class-number implications (e.g., in one scenario) and discuss how these results inform the study of -class field towers and cyclotomic -extensions related to the base biquadratic fields.

Abstract

Let and be two fields, where , and three different prime integers and be a positive odd square-free integer relatively prime to , and . The purpose of this paper is to show how one can proceed to perform the calculation of unit group of the fields of the form and . More precisely, we compute the unit group and the -class number of these fields whenever and or .
Paper Structure (4 sections, 14 theorems, 6 equations, 1 figure)

This paper contains 4 sections, 14 theorems, 6 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. The case: $p$, $q$ and $s$ satisfy conditions (\ref{['case1']})
  4. The case: $p$, $q$ and $s$ satisfy conditions (\ref{['case2']})

Key Result

Lemma 2.1

Let $K_0$ be an abelian field, $K=K_0(i)$ a quadratic extension of $K_0$, $n\geq 2$ an integer and $\xi_n$ a primitive $2^n$-th root of unity, then $\xi_n=\frac{1}{2}(\mu_n+\lambda_ni)$, where $\mu_n=\sqrt{2+\mu_{n-1}}$, $\lambda_n=\sqrt{2-\mu_{n-1}}$, $\mu_2=0$, $\lambda_2=2$ and $\mu_3=\lambda_3=\

Figures (1)

  • Figure 1: Subfields of $\mathbb{L}^+/\mathbb{Q}(\sqrt 2)$

Theorems & Definitions (25)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1: Ku-50
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • ...and 15 more