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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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R. Balasubramanian
Affiliation: Institute of Mathematical Sciences, C. I. T. Campus, Taramani, Chennai 600113, India

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

R. Thangadurai
Affiliation: Department of Mathematics, Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211019, India

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.

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