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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the exact degree of $ \mathbb{Q}(\sqrt{a_1}, \sqrt{a_2},\ldots, \sqrt{a_\ell})$ over $ \mathbb{Q}$


Authors: R. Balasubramanian, F. Luca and R. Thangadurai
Journal: Proc. Amer. Math. Soc. 138 (2010), 2283-2288
MSC (2010): Primary 11A15
Published electronically: March 15, 2010
MathSciNet review: 2607857
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Abstract: Let $ S =\{a_1, a_2, \ldots, a_\ell\}$ be a finite set of non-zero integers. In this paper, we give an exact formula for the degree of the multi-quadratic field $ \mathbb{Q}(\sqrt{a_1}, \sqrt{a_2},\ldots, \sqrt{a_\ell})$ over $ \mathbb{Q}$. To do this, we compute the relative density of the set of prime numbers $ p$ for which all the $ a_i$'s are simultaneously quadratic residues modulo $ p$ in two ways.


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Additional Information

R. Balasubramanian
Affiliation: Institute of Mathematical Sciences, C. I. T. Campus, Taramani, Chennai 600113, India
Email: balu@imsc.res.in

F. Luca
Affiliation: Mathematical Institute, Universidad Nacional Autónoma de México, Ap. Postal, 61-3 (Xangari), CP 58089, Morelia, Michoacán, Mexico
Email: fluca@matmor.unam.mx

R. Thangadurai
Affiliation: Department of Mathematics, Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211019, India
Email: thanga@hri.res.in

DOI: http://dx.doi.org/10.1090/S0002-9939-10-10331-1
PII: S 0002-9939(10)10331-1
Keywords: Quadratic residues, Galois field, Chebotarev density theorem
Received by editor(s): September 15, 2009
Published electronically: March 15, 2010
Communicated by: Ken Ono
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.