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Explicit methods for the Hasse norm principle and applications to An and Sn extensions

Published online by Cambridge University Press:  22 April 2021

ANDRÉ MACEDO
Affiliation:
University of Reading, Department of Mathematics and Statistics, Pepper Lane, Whiteknights, Reading RG6 6AX, UK e-mails: c.a.v.macedo@pgr.reading.ac.uk, r.d.newton@reading.ac.uk
RACHEL NEWTON
Affiliation:
University of Reading, Department of Mathematics and Statistics, Pepper Lane, Whiteknights, Reading RG6 6AX, UK e-mails: c.a.v.macedo@pgr.reading.ac.uk, r.d.newton@reading.ac.uk

Abstract

Let K/k be an extension of number fields. We describe theoretical results and computational methods for calculating the obstruction to the Hasse norm principle for K/k and the defect of weak approximation for the norm one torus \[R_{K/k}^1{\mathbb{G}_m}\] . We apply our techniques to give explicit and computable formulae for the obstruction to the Hasse norm principle and the defect of weak approximation when the normal closure of K/k has symmetric or alternating Galois group.

Type
Research Article
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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