Note on the Hilbert 2-class field tower

Abdelmalek Azizi; Mohamed Mahmoud Chems-Eddin; Abdelkader Zekhnini

Displaying similar documents to “Note on the Hilbert 2-class field tower”

The unit group of some fields of the form $ℚ (\sqrt{2}, \sqrt{p}, \sqrt{q}, \sqrt{- l})$

Moha Ben Taleb El Hamam (2024)

Mathematica Bohemica

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Let $p$ and $q$ be two different prime integers such that $p \equiv q \equiv 3 (mod 8)$ with $(p / q) = 1$ , and $l$ a positive odd square-free integer relatively prime to $p$ and $q$ . In this paper we investigate the unit groups of number fields $𝕃 = ℚ (\sqrt{2}, \sqrt{p}, \sqrt{q}, \sqrt{- l})$ .

On units of some fields of the form $ℚ (\sqrt{2}, \sqrt{p}, \sqrt{q}, \sqrt{- l})$

Mohamed Mahmoud Chems-Eddin (2023)

Mathematica Bohemica

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Let $p \equiv 1 (mod 8)$ and $q \equiv 3 (mod 8)$ be two prime integers and let $ℓ \notin {- 1, p, q}$ be a positive odd square-free integer. Assuming that the fundamental unit of $ℚ (\sqrt{2 p})$ has a negative norm, we investigate the unit group of the fields $ℚ (\sqrt{2}, \sqrt{p}, \sqrt{q}, \sqrt{- ℓ})$ .

Real quadratic number fields with metacyclic Hilbert $2$ -class field tower

Said Essahel, Ahmed Dakkak, Ali Mouhib (2019)

Mathematica Bohemica

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We begin by giving a criterion for a number field $K$ with 2-class group of rank 2 to have a metacyclic Hilbert 2-class field tower, and then we will determine all real quadratic number fields $ℚ (\sqrt{d})$ that have a metacyclic nonabelian Hilbert $2$ -class field tower.

On the Hilbert $2$ -class field tower of some imaginary biquadratic number fields

Mohamed Mahmoud Chems-Eddin, Abdelmalek Azizi, Abdelkader Zekhnini, Idriss Jerrari (2021)

Czechoslovak Mathematical Journal

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Let $𝕜 = ℚ (\sqrt{2}, \sqrt{d})$ be an imaginary bicyclic biquadratic number field, where $d$ is an odd negative square-free integer and $𝕜_{2}^{(2)}$ its second Hilbert $2$ -class field. Denote by $G = Gal (𝕜_{2}^{(2)} / 𝕜)$ the Galois group of $𝕜_{2}^{(2)} / 𝕜$ . The purpose of this note is to investigate the Hilbert $2$ -class field tower of $𝕜$ and then deduce the structure of $G$ .

Bicyclotomic polynomials and impossible intersections

David Masser, Umberto Zannier (2013)

Journal de Théorie des Nombres de Bordeaux

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In a recent paper we proved that there are at most finitely many complex numbers $t \neq 0, 1$ such that the points $(2, \sqrt{2 (2 - t)})$ and $(3, \sqrt{6 (3 - t)})$ are both torsion on the Legendre elliptic curve defined by $y^{2} = x (x - 1) (x - t)$ . In a sequel we gave a generalization to any two points with coordinates algebraic over the field $Q (t)$ and even over $C (t)$ . Here we reconsider the special case $(u, \sqrt{u (u - 1) (u - t)})$ and $(v, \sqrt{v (v - 1) (v - t)})$ with complex numbers $u$ and $v$ .

On the strongly ambiguous classes of some biquadratic number fields

Abdelmalek Azizi, Abdelkader Zekhnini, Mohammed Taous (2016)

Mathematica Bohemica

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We study the capitulation of $2$ -ideal classes of an infinite family of imaginary bicyclic biquadratic number fields consisting of fields $𝕜 = ℚ (\sqrt{2 p q}, i)$ , where $i = \sqrt{- 1}$ and $p \equiv - q \equiv 1 (mod 4)$ are different primes. For each of the three quadratic extensions $𝕂 / 𝕜$ inside the absolute genus field $𝕜^{(*)}$ of $𝕜$ , we determine a fundamental system of units and then compute the capitulation kernel of $𝕂 / 𝕜$ . The generators of the groups ${Am}_{s} (𝕜 / F)$ and $Am (𝕜 / F)$ are also determined from which we deduce that $𝕜^{(*)}$ is smaller than the relative genus field ${(𝕜 / ℚ (i))}^{*}$ . Then we prove...

On analytic rapidly decreasing functions of a real variable

Gianfranco Cimmino (1983)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Condizione necessaria e sufficiente affinché una funzione rapidamente decrescente di variabile reale sia uniformemente analitica è che per i suoi coefficienti $\gamma_{0},\gamma_{1},\dots$ di Fourier-Hermite riesca $\gamma_{m}=0(e^{\sqrt{mt}})$ per $t>0$ abbastanza piccolo.

Lower bound for class numbers of certain real quadratic fields

Mohit Mishra (2023)

Czechoslovak Mathematical Journal

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Let $d$ be a square-free positive integer and $h (d)$ be the class number of the real quadratic field $ℚ (\sqrt{d}) .$ We give an explicit lower bound for $h (n^{2} + r)$ , where $r = 1, 4$ . Ankeny and Chowla proved that if $g > 1$ is a natural number and $d = n^{2 g} + 1$ is a square-free integer, then $g ∣ h (d)$ whenever $n > 4$ . Applying our lower bounds, we show that there does not exist any natural number $n > 1$ such that $h (n^{2 g} + 1) = g$ . We also obtain a similar result for the family $ℚ (\sqrt{n^{2 g} + 4})$ . As another application, we deduce some criteria for a class group of prime power order to be...

Extension of CR functions to «wedge type» domains

Andrea D&amp;#039;Agnolo, Piero D&#039;Ancona, Giuseppe Zampieri (1991)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Let $X$ be a complex manifold, $S$ a generic submanifold of $X^{R}$ , the real underlying manifold to $X$ . Let $Ω$ be an open subset of $S$ with $\partial Ω$ analytic, $Y$ a complexification of $S$ . We first recall the notion of $Ω$ -tuboid of $X$ and of $Y$ and then give a relation between; we then give the corresponding result in terms of microfunctions at the boundary. We relate the regularity at the boundary for ${\bar{\partial}}_{b}$ to the extendability of $C R$ functions on $Ω$ to $Ω$ -tuboids of $X$ . Next, if $X$ has complex dimension 2, we give results...

A $C^{*}$ -Algebra Approach to Field Theory

D. Kastler (1968)

Recherche Coopérative sur Programme n°25

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