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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Étienne Fouvry
Affiliation: Laboratoire de Mathématiques d’Orsay, CNRS, Université Paris-Sud, F-91405 Orsay Cedex, France

Florian Luca
Affiliation: Instituto de Matemáticas, Universidad Nacional Autónoma de México, C.P. 58089, Morelia, Michoacán, México

Francesco Pappalardi
Affiliation: Dipartimento di Matematica, Università Roma Tre, Largo S. L. Murialdo, 1, Roma, 00146, Italy

Igor E. Shparlinski
Affiliation: Department of Computing, Macquarie University, Sydney, NSW 2109, Australia

Received by editor(s): September 21, 2009
Published electronically: January 11, 2011
Article copyright: © Copyright 2011 American Mathematical Society