Czechoslovak Mathematical Journal

Czechoslovak Mathematical Journal, Vol. 73, No. 3, pp. 937-940, 2023

On the class number of the maximal real subfields of a family of cyclotomic fields

Mahesh Kumar Ram

Received August 24, 2022. Published online April 27, 2023.

Abstract: For any square-free positive integer $m\equiv{10}\pmod{16}$ with $m\geq26$, we prove that the class number of the real cyclotomic field $\mathbb{Q}(\zeta_{4m}+\zeta_{4m}^{-1})$ is greater than $1$, where $\zeta_{4m}$ is a primitive $4m$th root of unity.

Keywords: maximal real subfield of cyclotomic field; real quadratic field; class number

Classification MSC: 11R29, 11R18, 11R11

DOI: 10.21136/CMJ.2023.0364-22

PDF available at: Springer Institute of Mathematics CAS Digital Mathematics Library

References:
[1] N. C. Ankeny, S. Chowla, H. Hasse: On the class-number of the maximal real subfield of a cyclotomic field. J. Reine Angew. Math. 217 (1965), 217-220. DOI 10.1515/crll.1965.217.217 | MR 172861 | Zbl 0128.03501
[2] A. Gica: Quadratic residues of certain types. Rocky Mt. J. Math. 36 (2006), 1867-1871. DOI 10.1216/rmjm/1181069349 | MR 2305634 | Zbl 1139.11004
[3] H. Hasse: Über mehrklassige, aber eingeschlechtige reell-quadratische Zahlkörper. Elem. Math. 20 (1965), 49-59. (In German.) MR 191889 | Zbl 0128.03502
[4] A. Hoque, K. Chakraborty: Pell-type equations and class number of the maximal real subfield of a cyclotomic field. Ramanujan J. 46 (2018), 727-742. DOI 10.1007/s11139-017-9963-9 | MR 3826752 | Zbl 1422.11065
[5] A. Hoque, H. K. Saikia: On the class-number of the maximal real subfield of a cyclotomic field. Quaest. Math. 39 (2016), 889-894. DOI 10.2989/16073606.2016.1188864 | MR 3573387 | Zbl 1423.11190
[6] S.-D. Lang: Note on the class-number of the maximal real subfield of a cyclotomic field. J. Reine Angew. Math. 290 (1977), 70-72. DOI 10.1515/crll.1977.290.70 | MR 447177 | Zbl 0346.12003
[7] H. Osada: Note on the class-number of the maximal real subfield of a cyclotomic field. Manuscr. Math. 58 (1987), 215-227. DOI 10.1007/BF01169091 | MR 884993 | Zbl 0602.12002
[8] H. Takeuchi: On the class-number of the maximal real subfield of a cyclotomic field. Can. J. Math. 33 (1981), 55-58. DOI 10.4153/CJM-1981-006-8 | MR 608854 | Zbl 0482.12004
[9] I. Yamaguchi: On the class-number of the maximal real subfield of a cyclotomic field. J. Reine Angew. Math. 272 (1975), 217-220. DOI 10.1515/crll.1975.272.217 | MR 366874 | Zbl 0313.12003

Affiliations: Mahesh Kumar Ram, Indian Institute of Science Education and Research, Govt. ITI Building Engg. School Junction, Brahmapur, Odisha 760010, India, e-mail: maheshk@iiserbpr.ac.in

Springer subscribers can access the papers on Springer website.

[List of online first articles] [Contents of Czechoslovak Mathematical Journal] [Full text of the older issues of Czechoslovak Mathematical Journal at DML-CZ]

PDF available at:

Springer
for subscribers

···

Institute of Mathematics CAS
fulltext not available (moving wall 24 months)

···

Digital Mathematics Library
fulltext not available (moving wall 24 months)