On the exact degree of $\mathbb {Q}(\sqrt {a_1}, \sqrt {a_2},\ldots , \sqrt {a_\ell })$ over $\mathbb {Q}$
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- by R. Balasubramanian, F. Luca and R. Thangadurai PDF
- Proc. Amer. Math. Soc. 138 (2010), 2283-2288 Request permission
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.References
- M. Fried, Arithmetical properties of value sets of polynomials, Acta Arith. 15 (1968/69), 91–115. MR 244150, DOI 10.4064/aa-15-2-91-115
- Christine S. Abel-Hollinger and Horst G. Zimmer, Torsion groups of elliptic curves with integral $j$-invariant over multiquadratic fields, Number-theoretic and algebraic methods in computer science (Moscow, 1993) World Sci. Publ., River Edge, NJ, 1995, pp. 69–87. MR 1377742
- Michael Laska and Martin Lorenz, Rational points on elliptic curves over $\textbf {Q}$ in elementary abelian $2$-extensions of $\textbf {Q}$, J. Reine Angew. Math. 355 (1985), 163–172. MR 772489, DOI 10.1515/crll.1985.355.163
- K. R. Matthews, A generalisation of Artin’s conjecture for primitive roots, Acta Arith. 29 (1976), no. 2, 113–146. MR 396448, DOI 10.4064/aa-29-2-113-146
- Steve Wright, Patterns of quadratic residues and nonresidues for infinitely many primes, J. Number Theory 123 (2007), no. 1, 120–132. MR 2295434, DOI 10.1016/j.jnt.2006.06.003
- S. Wright, A combinatorial problem related to quadratic non-residue modulo $p$, Ars Combinatorica, to appear.
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
- MR Author ID: 630217
- 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
- Received by editor(s): September 15, 2009
- Published electronically: March 15, 2010
- Communicated by: Ken Ono
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 2283-2288
- MSC (2010): Primary 11A15
- DOI: https://doi.org/10.1090/S0002-9939-10-10331-1
- MathSciNet review: 2607857