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# On the Content Bound for Real Quadratic Field Extensions

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 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.