On Tame Kernel and Class Group in Terms of Quadratic Forms

Yue, Qin

doi:10.1006/jnth.2002.2808

Journal of Number Theory

2002

DOI: 10.1006/jnth.2002.2808

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On Tame Kernel and Class Group in Terms of Quadratic Forms

Qin Yue¹

Abstract: The paper is to investigate the structure of the tame kernel K 2 O F for certain quadratic number fields F ; which extends the scope of Conner and Hurrelbrink (J. Number Theory 88 (2001), 263-282). We determine the 4-rank and the 8-rank of the tame kernel, the Tate kernel, and the 2-part of the class group. Our characterizations are in terms of binary quadratic formsThe results are very useful for numerical computations. # 2002 Elsevier Science (USA)

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Cited by 4 publications

References 13 publications

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The formula of 8-ranks of tame kernels

Yue

2005

Journal of Algebra

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In this paper, we always assume that F = Q( √ d) and E = Q( √ −d), d a squarefree integer, are quadratic number fields with d = u 2 − 2w 2 , u, w ∈ N. This paper is mainly to give the formula: 8-rank of K 2 O F = 8-rank of C(E) + a(F ) + σ , where C(E) is the narrow class group of E, a(F ) ∈ {−1, 0, 1} and σ ∈ {0, 1}; moreover |8-rank of K 2 O F -8-rank of C(E)| 1. This paper is also to show the relations among {−1, u + √ d} ∈ K 2 O 4 F , the dyadic ideal class of C(E) and the dyadic ideal class of C(E ) for E = Q( √ −2d).

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The formula of 8-ranks of tame kernels

Yue

2005

Journal of Algebra

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Tame kernels and further 4-rank densities

Osburn¹,

Murray²

2003

Journal of Number Theory

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There has been recent progress on computing the 4-rank of the tame kernel K 2 (O F ) for F a quadratic number field. For certain quadratic number fields, this progress has led to "density results" concerning the 4-rank of tame kernels. These results were first mentioned in [7] and proven in [9]. In this paper, we consider some additional quadratic number fields and obtain further density results of 4-ranks of tame kernels. Additionally, we give tables which might indicate densities in some generality. 257-274.

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The densities of 4-ranks of tame kernels for quadratic fields

Yue¹,

Jing²

2004

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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On Tame Kernel and Class Group in Terms of Quadratic Forms

Cited by 4 publications

References 13 publications

The formula of 8-ranks of tame kernels

The formula of 8-ranks of tame kernels

Tame kernels and further 4-rank densities

The densities of 4-ranks of tame kernels for quadratic fields

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