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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Counting dihedral and quaternionic extensions


Authors: Étienne Fouvry, Florian Luca, Francesco Pappalardi and Igor E. Shparlinski
Journal: Trans. Amer. Math. Soc. 363 (2011), 3233-3253
MSC (2010): Primary 11R11, 11R16; Secondary 11D09, 11L40
Published electronically: January 11, 2011
MathSciNet review: 2775805
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Abstract: We give asymptotic formulas for the number of biquadratic extensions of $ \mathbb{Q}$ that admit a quadratic extension which is a Galois extension of $ \mathbb{Q}$ with a prescribed Galois group, for example, with a Galois group isomorphic to the quaternionic group. Our approach is based on a combination of the theory of quadratic equations with some analytic tools such as the Siegel-Walfisz theorem and the double oscillations theorem.


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Additional Information

Étienne Fouvry
Affiliation: Laboratoire de Mathématiques d’Orsay, CNRS, Université Paris-Sud, F-91405 Orsay Cedex, France
Email: Etienne.Fouvry@math.u-psud.fr

Florian Luca
Affiliation: Instituto de Matemáticas, Universidad Nacional Autónoma de México, C.P. 58089, Morelia, Michoacán, México
Email: fluca@matmor.unam.mx

Francesco Pappalardi
Affiliation: Dipartimento di Matematica, Università Roma Tre, Largo S. L. Murialdo, 1, Roma, 00146, Italy
Email: pappa@mat.uniroma3.it

Igor E. Shparlinski
Affiliation: Department of Computing, Macquarie University, Sydney, NSW 2109, Australia
Email: igor.shparlinski@mq.edu.au

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05233-5
PII: S 0002-9947(2011)05233-5
Received by editor(s): September 21, 2009
Published electronically: January 11, 2011
Article copyright: © Copyright 2011 American Mathematical Society