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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Ratios of regulators in extensions of number fields


Authors: Antone Costa and Eduardo Friedman
Journal: Proc. Amer. Math. Soc. 119 (1993), 381-390
MSC: Primary 11R27; Secondary 11R29
MathSciNet review: 1152273
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Abstract: Let $ L/K$ be an extension of number fields. Then

$\displaystyle \operatorname{Reg} (L)/\operatorname{Reg} (K) > {c_{[L:{\mathbf{Q}}]}}{(\log \vert{D_L}\vert)^m},$

where Reg denotes the regulator, $ {D_L}$ is the absolute discriminant of $ L$, and $ {c_{[L:{\mathbf{Q}}]}} > 0$ depends only on the degree of $ L$. The nonnegative integer $ m = m(L/K)$ is positive if $ L/K$ does not belong to certain precisely defined infinite families of extensions, analogous to CM fields, along which $ \operatorname{Reg} (L)/\operatorname{Reg} (K)$ is constant. This generalizes some inequalities due to Remak and Silverman, who assumed that $ K$ is the rational field $ {\mathbf{Q}}$, and modifies those of Bergé-Martinet, who dealt with a general extension $ L/K$ but used its relative discriminant where we use the absolute one.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1152273-4
PII: S 0002-9939(1993)1152273-4
Keywords: Regulator, discriminant, unit-weak extensions
Article copyright: © Copyright 1993 American Mathematical Society