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On the Content Bound for Real Quadratic Field Extensions

by Robert G. Underwood
Department of Mathematics/Informatics Institute, Auburn University Montgomery, P. O. Box 244023, Montgomery, AL, USA
Axioms 2013, 2(1), 1-9; https://doi.org/10.3390/axioms2010001
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)
Let K be a finite extension of $ℚ$ 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 $c\text{}:\text{}{V}_{K}^{+}\to {ℝ}_{\ge 0}$ 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 cK 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 cK due to S. Lang. The purpose of this paper is to construct a new upper bound for cK 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. View Full-Text
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Underwood, R.G. On the Content Bound for Real Quadratic Field Extensions. Axioms 2013, 2, 1-9.

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