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Axioms 2013, 2(1), 19; doi:10.3390/axioms2010001
Article
On the Content Bound for Real Quadratic Field Extensions
Department of Mathematics/Informatics Institute, Auburn University Montgomery, P. O. Box 244023, Montgomery, AL, USA
Received: 31 October 2012 / Revised: 18 December 2012 / Accepted: 20 December 2012 / Published: 28 December 2012
(This article belongs to the Special Issue Axioms in Number Theory)
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Abstract
Let K be a finite extension of $\mathbb{Q}$ and let S = {ν} denote the collection of K normalized absolute values on K. Let ${V}_{K}^{+}$ denote the additive group of adeles over K and let $c\text{}:\text{}{V}_{K}^{+}\to {\mathbb{R}}_{\ge 0}$ denote the content map defined as $\text{c}\left(\left\{{\text{a}}_{\text{v}}\right\}\right)\text{=}{\prod}_{v\in s}v\left({\text{a}}_{\text{v}}\right)$ for $\left\{{a}_{v}\right\}\in {V}_{K}^{+}$. A classical result of J. W. S. Cassels states that there is a constant c > 0 depending only on the field K with the following property: if $\left\{{a}_{v}\right\}\in {V}_{K}^{+}$ with $c\left(\left\{{a}_{v}\right\}\right)\text{}>\text{}c$ , then there exists a nonzero element b ∈ K for which $v\left(b\right)\le v\left({a}_{v}\right),\text{}\forall v\in \text{}S$. Let c_{K} be the greatest lower bound of the set of all c that satisfy this property. In the case that K is a real quadratic extension there is a known upper bound for c_{K } due to S. Lang. The purpose of this paper is to construct a new upper bound for c_{K} in the case that K has class number one. We compare our new bound with Lang’s bound for various real quadratic extensions and find that our new bound is better than Lang’s in many instances.Keywords:
adele group; content map; real quadratic extension
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