Abstract
Let p be an odd prime and \(\ell \ge 1\) an integer. In this paper, we prove that the class numbers of \(\mathbb {Q}(\sqrt{p^{12\ell +2}-4})\) and \(\mathbb {Q}(\sqrt{(4-p^{12\ell }+2)/3})\) are divisible by 3. Using Siegel’s theorem, we show that there are infinitely many such pairs of quadratic fields with class number divisible by 3.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
N. Ankeny, E. Artin, S. Chowla, The class number of real quadratic fields. Ann. Math. 56, 479–493 (1952)
K. Chakraborty, A. Hoque, Y. Kishi, P.P. Pandey, Divisibility of the class numbers of imaginary quadratic fields. J. Number Theory 185, 339–348 (2018)
K. Chakraborty, A. Hoque, Class groups of imaginary quadratic fields of 3-rank at least 2. Ann. Univ. Sci. Budapest. Sect. Comput. 47, 179–183 (2018)
K. Chakraborty, A. Hoque, R. Sharma, Divisibility of class numbers of quadratic fields: Qualitative aspects, in Advances in Mathematical inequalities and Application (Trends Math, Birkhäuser/Springer, Singapore, 2018), pp. 247–264
K. Chakraborty, A. Hoque, Divisibility of class numbers of certain families of quadratic fields. J. Ramanujan Math. Soc. 34(3), 281–289 (2019)
K. Chakraborty, A. Hoque, Exponents of class groups of certain imaginary quadratic fields. Czechoslovak Math. J. (2019) (to appear)
J.-H. Evertse, J.H. Silverman, Uniform bounds for the number of solutions to \(Y^n = f(X)\). Math. Proc. Cambridge Philos. Soc. 100, 237–248 (1986)
A. Hoque, H.K. Saikia On generalized Mersenne primes and class-numbers of equivalent quadratic fields and cyclotomic fields. SeMA J. 67, 71–75 (2015)
A. Hoque, H.K. Saikia, A note on quadratic fields whose class numbers are divisible by \(3\). SeMA J. 73, 1–5 (2016)
H. Ichimura, Note on the class numbers of certain real quadratic fields. Abh. Math. Sem. Univ. Hamburg 73, 281–288 (2003)
A. Ito, Remarks on the divisibility of the class numbers of imaginary quadratic fields \(\mathbb{Q}(\sqrt{2^{2k}-q^n})\). Glasgow Math. J. 53, 379–389 (2011)
Y. Kishi, K. Miyake, parameterization of the quadratic fields whose class numbers are divisible by three. J. Number Theory 80, 209–217 (2000)
Y. Kishi, Note on the divisibility of the class number of certain imaginary quadratic fields. Glasgow Math. J. 51, 187–191 (2009)
S.R. Louboutin, On the divisibility of the class number of imaginary quadratic number fields. Proc. Amer. Math. Soc. 137, 4025–4028 (2009)
T. Nagell, Über die Klassenzahl imaginär quadratischer, Zählkörper. Abh. Math. Sem. Univ. Hamburg. 1, 140–150 (1922)
W.M. Schmidt, equations over finite fields, an elementary approach, in Lecture Notes in Math vol. 536 (Springer, Berlin, New York, 1976)
C.L. Siegel, Uber einige Anwendungen Diophantischer Approximationen, Abh. Preuss. Akad. Wiss. Phys. Math. Kl. 1, 1–70; Ges. Abh. 1, 209–266 (1929)
K. Soundararajan, Divisibility of class numbers of imaginary quadratic fields. J. Lond. Math. Soc. 61, 681–690 (2000)
K. Uchida, Unramified extension of quadratic number fields, II. Tohoku Math. J. 22, 220–224 (1970)
Y. Yamamoto, On unramified Galois extensions of quadratic number fields. Osaka J. Math. 7, 57–76 (1970)
P.J. Weinberger, Real Quadratic fields with Class number divisible by \(n\). J. Number theory 5, 237–241 (1973)
Acknowledgements
The authors would like to thank Azizul Hoque for suggesting the problem and also for a careful reading of this manuscript. The authors are also indebted to the anonymous for the thorough perusal of this paper.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Kalita, H., Saikia, H.K. (2020). A Pair of Quadratic Fields with Class Number Divisible by 3. In: Chakraborty, K., Hoque, A., Pandey, P. (eds) Class Groups of Number Fields and Related Topics. Springer, Singapore. https://doi.org/10.1007/978-981-15-1514-9_13
Download citation
DOI: https://doi.org/10.1007/978-981-15-1514-9_13
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-15-1513-2
Online ISBN: 978-981-15-1514-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)