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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Constructing abelian extensions with prescribed norms
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by Christopher Frei and Rodolphe Richard HTML | PDF
Math. Comp. 91 (2022), 381-399 Request permission

Abstract:

Given a number field $K$, a finite abelian group $G$ and finitely many elements $\alpha _1,\ldots ,\alpha _t\in K$, we construct abelian extensions $L/K$ with Galois group $G$ that realise all of the elements $\alpha _1,\ldots ,\alpha _t$ as norms of elements in $L$. In particular, this shows existence of such extensions for any given parameters.

Our approach relies on class field theory and a recent formulation of Tate’s characterisation of the Hasse norm principle, a local-global principle for norms. The constructions are sufficiently explicit to be implemented on a computer, and we illustrate them with concrete examples.

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Additional Information
  • Christopher Frei
  • Affiliation: TU Graz, Institute of Analysis and Number Theory, Steyrergasse 30/II, 8010 Graz, Austria
  • MR Author ID: 938397
  • ORCID: 0000-0001-8962-9240
  • Email: frei@math.tugraz.at
  • Rodolphe Richard
  • Affiliation: 13, rue du Croisic, 22200 Plouisy, Bretagne, France
  • Address at time of publication: University College London, 25 Gordon Street, WC1H 0AY London, United Kingdom
  • MR Author ID: 875874
  • Email: rodolphe.richard@normalesup.org
  • Received by editor(s): July 2, 2020
  • Received by editor(s) in revised form: December 14, 2020, and April 8, 2021
  • Published electronically: July 22, 2021
  • Additional Notes: The first author was supported by EPSRC grant EP/T01170X/1. The second author was supported by ERC grant GeTeMo 617129, and Leverhulme Research Project Grant “Diophantine problems related to Shimura varieties”
  • © Copyright 2021 American Mathematical Society
  • Journal: Math. Comp. 91 (2022), 381-399
  • MSC (2020): Primary 11Y40, 11R37; Secondary 14G05, 11D57
  • DOI: https://doi.org/10.1090/mcom/3663
  • MathSciNet review: 4350543