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# 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 non-zero element b ∈ K for which $v\left(b\right)\le v\left({a}_{v}\right),\text{}\forall v\in \text{}S$. Let

*c*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

_{K}*c*due to S. Lang. The purpose of this paper is to construct a new upper bound for

_{K }*c*in the case that

_{K}*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

*This is an open access article distributed under the Creative Commons Attribution License (CC BY) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.*

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**MDPI and ACS Style**

Underwood, R.G. On the Content Bound for Real Quadratic Field Extensions. *Axioms* **2013**, *2*, 1-9.