Hilbert symbols, class groups and quaternion algebras

Ted Chinburg; Eduardo Friedman

Displaying similar documents to “Hilbert symbols, class groups and quaternion algebras”

Note on the Hilbert 2-class field tower

Abdelmalek Azizi, Mohamed Mahmoud Chems-Eddin, Abdelkader Zekhnini (2022)

Mathematica Bohemica

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Let $k$ be a number field with a 2-class group isomorphic to the Klein four-group. The aim of this paper is to give a characterization of capitulation types using group properties. Furthermore, as applications, we determine the structure of the second 2-class groups of some special Dirichlet fields $𝕜 = ℚ (\sqrt{d}, \sqrt{- 1})$ , which leads to a correction of some parts in the main results of A. Azizi and A. Zekhini (2020).

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 the ring of $p$ -integers of a cyclic $p$ -extension over a number field

Humio Ichimura (2005)

Journal de Théorie des Nombres de Bordeaux

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Let $p$ be a prime number. A finite Galois extension $N / F$ of a number field $F$ with group $G$ has a normal $p$ -integral basis ( $p$ -NIB for short) when $𝒪_{N}^{'}$ is free of rank one over the group ring $𝒪_{F}^{'} [G]$ . Here, $𝒪_{F}^{'} = 𝒪_{F} [1 / p]$ is the ring of $p$ -integers of $F$ . Let $m = p^{e}$ be a power of $p$ and $N / F$ a cyclic extension of degree $m$ . When $ζ_{m} \in F^{\times}$ , we give a necessary and sufficient condition for $N / F$ to have a $p$ -NIB (Theorem 3). When $ζ_{m} \notin F^{\times}$ and $p ∤ [F (ζ_{m}) : F]$ , we show that $N / F$ has a $p$ -NIB if and only if $N (ζ_{m}) / F (ζ_{m})$ has a $p$ -NIB (Theorem 1). When $p$ divides $[F (ζ_{m}) : F]$ , we show that this...

Exponent of class group of certain imaginary quadratic fields

Kalyan Chakraborty, Azizul Hoque (2020)

Czechoslovak Mathematical Journal

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Let $n > 1$ be an odd integer. We prove that there are infinitely many imaginary quadratic fields of the form $ℚ (\sqrt{x^{2} - 2 y^{n}})$ whose ideal class group has an element of order $n$ . This family gives a counterexample to a conjecture by H. Wada (1970) on the structure of ideal class groups.

A new characterization of Suzuki groups

Behnam Ebrahimzadeh, Reza Mohammadyari (2019)

Archivum Mathematicum

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One of the important questions that remains after the classification of the finite simple groups is how to recognize a simple group via specific properties. For example, authors have been able to use graphs associated to element orders and to number of elements with specific orders to determine simple groups up to isomorphism. In this paper, we prove that Suzuki groups $S z (q)$ , where $q \pm \sqrt{2 q} + 1$ is a prime number can be uniquely determined by the order of group and the number of elements with the same...

On some metabelian 2-groups and applications I

Abdelmalek Azizi, Abdelkader Zekhnini, Mohammed Taous (2016)

Colloquium Mathematicae

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Let G be some metabelian 2-group satisfying the condition G/G’ ≃ ℤ/2ℤ × ℤ/2ℤ × ℤ/2ℤ. In this paper, we construct all the subgroups of G of index 2 or 4, we give the abelianization types of these subgroups and we compute the kernel of the transfer map. Then we apply these results to study the capitulation problem for the 2-ideal classes of some fields k satisfying the condition $G a l (k ₂^{(2)} / k) ≃ G$ , where $k ₂^{(2)}$ is the second Hilbert 2-class field of k.

On the ideal class groups of the maximal real subfields of number fields with all roots of unity

Masato Kurihara (1999)

Journal of the European Mathematical Society

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In this paper, for a totally real number field $k$ we show the ideal class group of $k {(\cup_{n > 0} μ_{n})}^{+}$ is trivial. We also study the $p$ -component of the ideal class group of the cyclotomic $𝐙_{p}$ -extension.

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 $G$ .

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.