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Redei reciprocity, governing fields and negative Pell

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  19 April 2021

PETER STEVENHAGEN
PETER STEVENHAGEN*
Affiliation:
Mathematisch Instituut, Universiteit Leiden, Postbus 9512, 2300 RA Leiden, The Netherlands e-mail: psh@math.leidenuniv.nl
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Abstract

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We discuss the origin, an improved definition and the key reciprocity property of the trilinear symbol introduced by Rédei [16] in the study of 8-ranks of narrow class groups of quadratic number fields. It can be used to show that such 8-ranks are ‘governed’ by Frobenius conditions on the primes dividing the discriminant, a fact used in the recent work of A. Smith [18, 19]. In addition, we explain its impact in the progress towards proving my conjectural density for solvability of the negative Pell equation $x^2-dy^2=-1$ .

MSC classification

Primary: 11R11: Quadratic extensions 11R37: Class field theory
Secondary: 11R16: Cubic and quartic extensions
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 172 , Issue 3 , May 2022 , pp. 627 - 654
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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