On the Stark-Shintani units and the ideal class groups in the cyclotomic $ℤ_p$-extensions of class fields over real quadratic fields

Tsuyoshi Itoh

Displaying similar documents to “On the Stark-Shintani units and the ideal class groups in the cyclotomic $ℤ_{p}$ -extensions of class fields over real quadratic fields”

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.

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.

Class group of a cyclotomic $ℤ_{p} \times ℤ_{ℓ}$ -extension

Humio Ichimura (2011)

Acta Arithmetica

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

Christian Maire, Cam McLeman (2014)

Annales mathématiques Blaise Pascal

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Much recent progress in the 2-class field tower problem revolves around demonstrating infinite such towers for fields – in particular, quadratic fields – whose class groups have large 4-ranks. Generalizing to all primes, we use Golod-Safarevic-type inequalities to analyse the source of the $p^{2}$ -rank of the class group as a quantity of relevance in the $p$ -class field tower problem. We also make significant partial progress toward demonstrating that all real quadratic number fields whose class...

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...

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.

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...

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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$H^{p}$ -spaces of conjugate systems on local fields

Jia Chao (1974)

Studia Mathematica

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On the subfields of cyclotomic function fields

Zhengjun Zhao, Xia Wu (2013)

Czechoslovak Mathematical Journal

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Let $K = 𝔽_{q} (T)$ be the rational function field over a finite field of $q$ elements. For any polynomial $f (T) \in 𝔽_{q} [T]$ with positive degree, denote by $Λ_{f}$ the torsion points of the Carlitz module for the polynomial ring $𝔽_{q} [T]$ . In this short paper, we will determine an explicit formula for the analytic class number for the unique subfield $M$ of the cyclotomic function field $K (Λ_{P})$ of degree $k$ over $𝔽_{q} (T)$ , where $P \in 𝔽_{q} [T]$ is an irreducible polynomial of positive degree and $k > 1$ is a positive divisor of $q - 1$ . A formula for the analytic class...

Displaying similar documents to “On the Stark-Shintani units and the ideal class groups in the cyclotomic ℤ p -extensions of class fields over real quadratic fields”

Displaying similar documents to “On the Stark-Shintani units and the ideal class groups in the cyclotomic $ℤ_{p}$ -extensions of class fields over real quadratic fields”