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Prof. Dr. Mirela Stefanescu

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Faculty of Sciences, Ovidius University of Constanza, Mamaia 124, 900527 Constanta, Romania
Interests: axiomatic theories in algebra; group theory; field and ring theory; number theory; axiomatic methods; axiomatization

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172 KiB

Open AccessArticle

On the Content Bound for Real Quadratic Field Extensions

by Robert G. Underwood

Axioms 2013, 2(1), 1-9; https://doi.org/10.3390/axioms2010001 - 28 Dec 2012

Viewed by 4356

Abstract

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 [...] Read more.

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 : V_{K}^{+} \to ℝ_{\geq 0}

denote the content map defined as

c ({a_{v}}) = \prod_{v \in s} v (a_{v})

for

{a_{v}} \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

{a_{v}} \in V_{K}^{+}

with

c ({a_{v}}) > c

, then there exists a non-zero element b ∈ K for which

v (b) \leq v (a_{v}), \forall v \in 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. Full article

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