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Weil numbers in finite extensions of $ \mathbb{Q}^{ab}$: the Loxton-Kedlaya phenomenon


Authors: Florin Stan and Alexandru Zaharescu; with an appendix by Kiran S. Kedlaya
Journal: Trans. Amer. Math. Soc. 367 (2015), 4359-4376
MSC (2010): Primary 11R18, 11R06
DOI: https://doi.org/10.1090/S0002-9947-2014-06414-3
Published electronically: November 18, 2014
MathSciNet review: 3324931
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Abstract: A finiteness phenomenon described by Loxton and later by Kedlaya states that, for any fixed $ m$, there exist (modulo multiplication by roots of unity) only finitely many $ m$-Weil numbers in $ \mathbb{Q}^{ab}$. In the present paper we show that this phenomenon extends to all finite extensions of $ \mathbb{Q}^{ab}$.


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Additional Information

Florin Stan
Affiliation: Simion Stoilow Institute of Mathematics of the Romanian Academy, Research unit 5, PO Box 1-764, RO-014700 Bucharest, Romania
Email: sfloringabriel@yahoo.com

Alexandru Zaharescu
Affiliation: Simion Stoilow Institute of Mathematics of the Romanian Academy, Research unit 5, PO Box 1-764, RO-014700 Bucharest, Romania — and — Department of Mathematics, University of Illinois at Urbana-Champaign, Altgeld Hall, 1409 W. Green Street, Urbana, Illinois 61801
Email: zaharesc@illinois.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-06414-3
Keywords: Weil numbers, cyclotomic fields
Received by editor(s): September 18, 2012
Received by editor(s) in revised form: August 21, 2013
Published electronically: November 18, 2014
Additional Notes: The research of the second author was supported by the NSF grant DMS-0901621.
Article copyright: © Copyright 2014 American Mathematical Society