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Arithmetic of some real triquadratic fields; Units and 2-class groups

Mohamed Mahmoud Chems-Eddin

Abstract

In this paper, we compute the unit groups and the $2$-class numbers of the Fröhlich's triquadratic fields $\KK=\mathbb{Q}(\sqrt{2},\sqrt{p},\sqrt{q})$, where $p$ and $q$ are two prime numbers such that ($p\equiv 1 \pmod8$ and $q\equiv 3 \pmod4$) or ($p\equiv 5$ or $3 \pmod8$ and $q\equiv 3 \pmod4$). Furthermore, we determine some families of the fields $\KK$ whose $2$-class groups are trivial or cyclic non trivial, and some other families with $2$-class groups isomorphic to the Klein group.

Arithmetic of some real triquadratic fields; Units and 2-class groups

Abstract

In this paper, we compute the unit groups and the -class numbers of the Fröhlich's triquadratic fields , where and are two prime numbers such that ( and ) or ( or and ). Furthermore, we determine some families of the fields whose -class groups are trivial or cyclic non trivial, and some other families with -class groups isomorphic to the Klein group.

Paper Structure

This paper contains 7 sections, 20 theorems, 6 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Preparations
  3. Unit groups of real triquadratic number fields and their $2$-class numbers
  4. The case: $p\equiv 1 \pmod8$, $q\equiv 3 \pmod4$ and $\genfrac(){}{0}{p}{q} =-1$
  5. The case: $p\equiv 1 \pmod8$, $q\equiv 3 \pmod8$ and $\genfrac(){}{0}{p}{q} =1$
  6. The cases: $p\equiv 5$ or $3 \pmod8$ and $q\equiv 3 \pmod4$
  7. Some families of Fröhlich triquadratic fields whose $2$-class groups are of type $(2,2)$

Key Result

Lemma 2.1

Let $p$ be a prime number such that $N(\varepsilon_{2p})=1$. Put $\varepsilon_{2p}=\beta+\alpha\sqrt{2p}$ with $\beta , \alpha\in\mathbb{Z}$. Then $\sqrt{\varepsilon_{2p}}=\frac{1}{\sqrt2}(\alpha_1+\alpha_2\sqrt{ 2p})$, for some integers $\alpha_1, \alpha_2$ such that $\alpha=\alpha_1 \alpha_2$. It for some $u$ in $\{0, 1\}$ such that $\frac{1}{2}(\alpha_1^2-2p\alpha_2^2)=(-1)^u$. With ${\varepsi

Figures (1)

  • Figure 1: Intermediate fields of $\mathbb{K}/\mathbb{Q}(\sqrt 2)$

Theorems & Definitions (36)

  • Lemma 2.1
  • Lemma 2.2: Az-00, Lemma 5
  • proof : Proof of Lemma \ref{['lm noms esp_2p']}
  • Lemma 2.3: azizitalbi, Theorem 6
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 26 more