On the Plus Parts of the Class Numbers of Cyclotomic Fields

Abstract

The authors exhibit some new families of cyclotomic fields which have non-trivial plus parts of their class numbers. They also prove the 3-divisibility of the plus part of the class number of another family consisting of infinitely many cyclotomic fields. At the end, they provide some numerical examples supporting our results.

Keywords

Class numbers / Maximal real subfield of cyclotomic fields / Real quadratic fields

References

[1] AnkenyN C, ChowlaS, HasseH. On the class-number of the maximal real subfield of a cyclotomic field. J. Reine Angew. Math., 1965, 217: 217-220

[2] BuhlerJ P, HarveyD. Irregular primes to 163 million. Math. Comp., 2011, 80: 2435-2444

[3] ChakrabortyK, HoqueA, SharmaRAgarwalP, DragomirS, JleliM, SametB. Divisibility of Class Numbers of Quadratic Fields: Qualitative Aspects. Advances In Mathematical inequalities and Application, 2018, Singapore, Birkhäuser, Springer-Verlag: 247-264

[4] HoqueA, ChakrabortyK. Pell-type equations and class number of the maximal real subfield of a cyclotomic field. Ramanujan J., 2018, 46(3): 727-742

[5] HoqueA, ChakrabortyK. Divisibility of class numbers of certain families of quadratic fields. J. Ramanujan Math. Soc., 2019, 34(3): 281-289

[6] HoqueA, SaikiaH K. On the class-number of the maximal real subfield of a cyclotomic field. Quaest. Math., 2016, 39(7): 889-894

[7] KishiY. A constructive approach to Spiegelung relations between 3-Ranks of absolute ideal class groups and congruent ones modulo (3)² in quadratic fields. J. Number Theory, 2000, 83: 1-49

[8] KummerE E. Bestimmung der Anzahl nicht äquivalenter Classen für die aus λ^ten Wurzeln der Einheit gebildeten complexen Zahlen und die idealen Factoren derselben. J. Reine Angew. Math., 1850, 40: 93-116

[9] KummerE E. Mémoire sur la théorie des nombres complexes composés de racines de l’unité et de nombres entiers. J. de math. pures et appl., 1851, 16: 377-498

[10] LangSCyclotomic Fields, 1978, New York, Springer-Verlag59

[11] LangS D. Note on the class-number of the maximal real subfield of a cyclotomic field. J. Reine Angew. Math., 1977, 290: 70-72

[12] LlorenteP, NartE. Effective determination of the decomposition of the rational prime in a cubic field. Proc. Amer. Math. Soc., 1983, 87: 579-585

[13] MillerJ C. Real cyclotomic fields of prime conductor and their class numbers. Math. Comp., 2015, 84: 2459-2469

[14] MollinR A. Diophantine equations and class numbers. J. Number Theory, 1986, 24: 7-19

[15] OsadaH. Note on the class-number of the maximal real subfield of a cyclotomic field. Manuscripta Math., 1987, 58: 215-227

[16] OsadaH. Note on the class-number of the maximal real subfield of a cyclotomic field, II. Nagoya Math. J., 1989, 113: 147-151

[17] The PARI GroupPARI/GP version 2.9.1, 2017

[18] SchoofR. Minus class groups of the fields of the l-th roots of unity. Math. Comp., 1998, 67: 1225-1245

[19] Schrutka von RechtenstammG. Tabelle der (Relativ)-Klassenzahlen der Kreiskörper, deren ϕ-Funktion des Wurzelexponenten (Grad) nicht grösser als 256 ist. Abh. Deutsch. Akad. Wiss. Berlin Kl. Math. Phys. Tech., 1964, 2: 1-64

[20] TakeuchiH. On the class-number of the maximal real subfield of a cyclotomic field. Can. J. Math., 1981, 33(1): 55-58

RIGHTS & PERMISSIONS

The Editorial Office of CAM and Springer-Verlag Berlin Heidelberg