Counting dihedral and quaternionic extensions
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- by Étienne Fouvry, Florian Luca, Francesco Pappalardi and Igor E. Shparlinski PDF
- Trans. Amer. Math. Soc. 363 (2011), 3233-3253 Request permission
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.References
Additional Information
- Étienne Fouvry
- Affiliation: Laboratoire de Mathématiques d’Orsay, CNRS, Université Paris-Sud, F-91405 Orsay Cedex, France
- ORCID: 0000-0002-1840-9467
- Email: Etienne.Fouvry@math.u-psud.fr
- Florian Luca
- Affiliation: Instituto de Matemáticas, Universidad Nacional Autónoma de México, C.P. 58089, Morelia, Michoacán, México
- MR Author ID: 630217
- 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
- MR Author ID: 192194
- Email: igor.shparlinski@mq.edu.au
- Received by editor(s): September 21, 2009
- Published electronically: January 11, 2011
- © Copyright 2011 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 363 (2011), 3233-3253
- MSC (2010): Primary 11R11, 11R16; Secondary 11D09, 11L40
- DOI: https://doi.org/10.1090/S0002-9947-2011-05233-5
- MathSciNet review: 2775805