On $p^2$-Ranks in the Class Field Tower Problem

Christian Maire; Cam McLeman

Displaying similar documents to “On $p^{2}$ -Ranks in the Class Field Tower Problem”

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.

Class groups of large ranks in biquadratic fields

Mahesh Kumar Ram (2024)

Czechoslovak Mathematical Journal

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For any integer $n > 1$ , we provide a parametric family of biquadratic fields with class groups having $n$ -rank at least 2. Moreover, in some cases, the $n$ -rank is bigger than 4.

Subfields of henselian valued fields

Ramneek Khassa, Sudesh K. Khanduja (2010)

Colloquium Mathematicae

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Let (K,v) be a henselian valued field of arbitrary rank which is not separably closed. Let k be a subfield of K of finite codimension and $v_{k}$ be the valuation obtained by restricting v to k. We give some necessary and sufficient conditions for $(k, v_{k})$ to be henselian. In particular, it is shown that if k is dense in its henselization, then $(k, v_{k})$ is henselian. We deduce some well known results proved in this direction through other considerations.

On the structure of the Galois group of the Abelian closure of a number field

Georges Gras (2014)

Journal de Théorie des Nombres de Bordeaux

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From a paper by A. Angelakis and P. Stevenhagen on the determination of a family of imaginary quadratic fields $K$ having isomorphic absolute Abelian Galois groups $A_{K}$ , we study any such issue for arbitrary number fields $K$ . We show that this kind of property is probably not easily generalizable, apart from imaginary quadratic fields, because of some $p$ -adic obstructions coming from the global units of $K$ . By restriction to the $p$ -Sylow subgroups of $A_{K}$ and assuming the Leopoldt conjecture we...

Another look at real quadratic fields of relative class number 1

Debopam Chakraborty, Anupam Saikia (2014)

Acta Arithmetica

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The relative class number $H_{d} (f)$ of a real quadratic field K = ℚ (√m) of discriminant d is defined to be the ratio of the class numbers of $_{f}$ and $_{K}$ , where $_{K}$ denotes the ring of integers of K and $_{f}$ is the order of conductor f given by $ℤ + f_{K}$ . R. Mollin has shown recently that almost all real quadratic fields have relative class number 1 for some conductor. In this paper we give a characterization of real quadratic fields with relative class number 1 through an elementary approach considering the...

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

D. Kastler (1968)

Recherche Coopérative sur Programme n°25

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The distribution of second $p$ -class groups on coclass graphs

Daniel C. Mayer (2013)

Journal de Théorie des Nombres de Bordeaux

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General concepts and strategies are developed for identifying the isomorphism type of the second $p$ -class group $G = Gal (F_{p}^{2} (K) | K)$ , that is the Galois group of the second Hilbert $p$ -class field $F_{p}^{2} (K)$ , of a number field $K$ , for a prime $p$ . The isomorphism type determines the position of $G$ on one of the coclass graphs $𝒢 (p, r)$ , $r \geq 0$ , in the sense of Eick, Leedham-Green, and Newman. It is shown that, for special types of the base field $K$ and of its $p$ -class group ${Cl}_{p} (K)$ , the position of $G$ is restricted to certain admissible branches...

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...

Governing fields for the 2-classgroup of $Q (s q r t - q_{1} q_{2} p)$ and a related reciprocity law

Patrick Morton (1990)

Acta Arithmetica

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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 $G$ .

Weighted Erdős-Kac type theorem over quadratic field in short intervals

Xiaoli Liu, Zhishan Yang (2022)

Czechoslovak Mathematical Journal

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Let $𝕂$ be a quadratic field over the rational field and $a_{𝕂} (n)$ be the number of nonzero integral ideals with norm $n$ . We establish Erdős-Kac type theorems weighted by $a_{𝕂} {(n)}^{l}$ and $a_{𝕂} {(n^{2})}^{l}$ of quadratic field in short intervals with $l \in ℤ^{+}$ . We also get asymptotic formulae for the average behavior of $a_{𝕂} {(n)}^{l}$ and $a_{𝕂} {(n^{2})}^{l}$ in short intervals.

Bicyclic commutator quotients with one non-elementary component

Daniel Mayer (2023)

Mathematica Bohemica

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For any number field $K$ with non-elementary $3$ -class group ${Cl}_{3} (K) ≃ C_{3^{e}} \times C_{3}$ , $e \geq 2$ , the punctured capitulation type $ϰ (K)$ of $K$ in its unramified cyclic cubic extensions $L_{i}$ , $1 \leq i \leq 4$ , is an orbit under the action of $S_{3} \times S_{3}$ . By means of Artin’s reciprocity law, the arithmetical invariant $ϰ (K)$ is translated to the punctured transfer kernel type $ϰ (G_{2})$ of the automorphism group $G_{2} = Gal (F_{3}^{2} (K) / K)$ of the second Hilbert $3$ -class field of $K$ . A classification of finite $3$ -groups $G$ with low order and bicyclic commutator quotient $G / G^{'} ≃ C_{3^{e}} \times C_{3}$ , $2 \leq e \leq 6$ , according to the algebraic...

Principalization algorithm via class group structure

Daniel C. Mayer (2014)

Journal de Théorie des Nombres de Bordeaux

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For an algebraic number field $K$ with $3$ -class group ${Cl}_{3} (K)$ of type $(3, 3)$ , the structure of the $3$ -class groups ${Cl}_{3} (N_{i})$ of the four unramified cyclic cubic extension fields $N_{i}$ , $1 \leq i \leq 4$ , of $K$ is calculated with the aid of presentations for the metabelian Galois group $G_{3}^{2} (K) = Gal (F_{3}^{2} (K) | K)$ of the second Hilbert $3$ -class field $F_{3}^{2} (K)$ of $K$ . In the case of a quadratic base field $K = ℚ (\sqrt{D})$ it is shown that the structure of the $3$ -class groups of the four $S_{3}$ -fields $N_{1}, ..., N_{4}$ frequently determines the type of principalization of the $3$ -class group of $K$ in $N_{1}, ..., N_{4}$ . This...

Displaying similar documents to “On p 2 -Ranks in the Class Field Tower Problem”

Displaying similar documents to “On $p^{2}$ -Ranks in the Class Field Tower Problem”