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Keywords:
quadratic field; discriminant; class group; Wada's conjecture
quadratic field; discriminant; class group; Wada's conjecture
Summary:
Let $n>1$ be an odd integer. We prove that there are infinitely many imaginary quadratic fields of the form $\mathbb {Q} \bigl (\sqrt {x^2-2y^n} \bigr )$ whose ideal class group has an element of order $n$. This family gives a counterexample to a conjecture by H. Wada (1970) on the structure of ideal class groups.
References:
[1] Ankeny, N. C., Chowla, S.: On the divisibility of the class number of quadratic fields. Pac. J. Math. 5 (1955), 321-324. DOI 10.2140/pjm.1955.5.321 | MR 0085301 | Zbl 0065.02402
[2] Chakraborty, K., Hoque, A.: Class groups of imaginary quadratic fields of 3-rank at least 2. Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Comput. 47 (2018), 179-183. MR 3849201 | Zbl 1424.11157
[3] Chakraborty, K., Hoque, A., Kishi, Y., Pandey, P. P.: Divisibility of the class numbers of imaginary quadratic fields. J. Number Theory 185 (2018), 339-348. DOI 10.1016/j.jnt.2017.09.007 | MR 3734353 | Zbl 06820888
[4] Chakraborty, K., Hoque, A., Sharma, R.: Divisibility of class numbers of quadratic fields: Qualitative aspects. Advances in Mathematical Inequalities and Applications Trends in Mathematics. Birkhäuser, Singapore (2018), 247-264. DOI 10.1007/978-981-13-3013-1_12 | MR 3969663 | Zbl 1418.11144
[5] Cohn, J. H. E.: On the class number of certain imaginary quadratic fields. Proc. Am. Math. Soc. 130 (2002), 1275-1277. DOI 10.1090/S0002-9939-01-06255-4 | MR 1879947 | Zbl 1113.11063
[6] Gross, B. H., Rohrlich, D. E.: Some results on the Mordell-Weil group of the Jacobian of the Fermat curve. Invent. Math. 44 (1978), 201-224. DOI 10.1007/BF01403161 | MR 491708 | Zbl 0369.14011
[7] Hoque, A., Chakraborty, K.: Divisibility of class numbers of certain families of quadratic fields. J. Ramanujan Math. Soc. 34 (2019), 281-289. MR 4010381
[8] Hoque, A., Saikia, H. K.: On the divisibility of class numbers of quadratic fields and the solvability of Diophantine equations. SeMA J. 73 (2016), 213-217. DOI 10.1007/s40324-016-0065-1 | MR 3542827 | Zbl 1348.11087
[9] Ishii, K.: On the divisibility of the class number of imaginary quadratic fields. Proc. Japan Acad., Ser. A 87 (2011), 142-143. DOI 10.3792/pjaa.87.142 | MR 2843095 | Zbl 1262.11093
[10] Ito, A.: A note on the divisibility of class numbers of imaginary quadratic fields $\mathbb{Q}(\sqrt{a^2-k^n})$. Proc. Japan Acad., Ser. A 87 (2011), 151-155. DOI 10.3792/pjaa.87.151 | MR 2863357 | Zbl 1247.11139
[11] Ito, A.: Remarks on the divisibility of the class numbers of imaginary quadratic fields $\mathbb{Q}(\sqrt{2^{2k}-q^n})$. Glasg. Math. J. 53 (2011), 379-389. DOI 10.1017/S0017089511000012 | MR 2783167 | Zbl 1259.11102
[12] Ito, A.: Notes on the divisibility of the class numbers of imaginary quadratic fields $\mathbb{Q}(\sqrt{3^{2e}-4k^n})$. Abh. Math. Semin. Univ. Hamb. 85 (2015), 1-21. DOI 10.1007/s12188-015-0106-1 | MR 3334456 | Zbl 1400.11145
[13] Kishi, Y.: Note on the divisibility of the class number of certain imaginary quadratic fields. Glasg. Math. J. 51 (2009), 187-191 corrigendum ibid. 52 2010 207-208. DOI 10.1017/S001708950800462X | MR 2471686 | Zbl 1211.11124
[14] Louboutin, S. R.: On the divisibility of the class number of imaginary quadratic number fields. Proc. Am. Math. Soc. 137 (2009), 4025-4028. DOI 10.1090/S0002-9939-09-10021-7 | MR 2538563 | Zbl 1269.11111
[15] Murty, M. R.: The $abc$ conjecture and exponents of class groups of quadratic fields. Number Theory. Proceedings of the International Conference on Discrete Mathematics and Number Theory Contemporary Mathematics 210. American Mathematical Society, Providence (1998), 85-95. DOI 10.1090/conm/210 | MR 1478486 | Zbl 0893.11043
[16] Murty, M. R.: Exponents of class groups of quadratic fields. Topics in Number Theory Mathematics and Its Applications 467. Kluwer Academic Publishers, Dordrecht (1999), 229-239. DOI 10.1007/978-1-4613-0305-3_15 | MR 1691322 | Zbl 0993.11059
[17] Nagell, T.: Über die Klassenzahl imaginär-quadratischer Zählkörper. Abh. Math. Semin. Univ. Hamb. 1 (1922), 140-150 German \99999JFM99999 48.0170.03. DOI 10.1007/BF02940586 | MR 3069394
[18] Soundararajan, K.: Divisibility of class numbers of imaginary quadratic fields. J. Lond. Math. Soc., II. Ser. 61 (2000), 681-690. DOI 10.1112/S0024610700008887 | MR 1766097 | Zbl 1018.11054
[19] Group, The PARI: PARI/GP, version 2.9.0. University Bordeaux (2016), Available at http://pari.math.u-bordeaux.fr
[20] Wada, H.: A table of ideal class groups of imaginary quadratic fields. Proc. Japan Acad. 46 (1970), 401-403. DOI 10.3792/pja/1195520300 | MR 0366866 | Zbl 0209.35604
[21] Zhu, M., Wang, T.: The divisibility of the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{2^{2m}-k^n})$. Glasg. Math. J. 54 (2012), 149-154. DOI 10.1017/S0017089511000486 | MR 2862392 | Zbl 1269.11107
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