Relative class numbers inside the \(p\)th cyclotomic field. (English) Zbl 1458.11157

Any prime number \(p\equiv 3\pmod{4}\) can be written (not uniquely) in the form \(p=2nl^f+1\) for some odd \(n\) and prime \(l\) with \(l\nmid n\). Now, for every \(0\leq t\leq f\) we can define \(K_t\) the imaginary subfield of \(\mathbb{Q}(\zeta_p)\) of degree \(t\) and let \(h_t^{-}\) the relative class number of \(K_t\). In this paper, the authors give some divisibility results about the the ratio \(h_t^{-}/h_{t-1}^{-}\).


11R29 Class numbers, class groups, discriminants
11R18 Cyclotomic extensions


Full Text: Euclid


[1] T. Agoh: On the relative class number of special cyclotomic fields, Math. Appl. 1 (2012), 1-12. Zentralblatt MATH: 1286.11173
Digital Object Identifier: doi:10.13164/ma.2012.01
· Zbl 1286.11173
[2] P. Cornacchia: The parity of the class number of the cyclotomic fields of prime conductor, Proc. Amer. Math. Soc. 125 (1997), 3163-3168. Zentralblatt MATH: 1036.11509
Digital Object Identifier: doi:10.1090/S0002-9939-97-03909-9
· Zbl 1036.11509
[3] D. Davis: Computing the number of totally positive circular units which are squares, J. Number Theory 10 (1978), 1-9. · Zbl 0369.12002
[4] D.R. Estes: On the parity of the class number of the field of \(q\) th roots of unity, Rocky Mountain J. Math. 19 (1989), 675-682. · Zbl 0703.11052
[5] S. Fujima and H. Ichimura: Note on the class number of the \(p\) th cyclotomic field, Funct. Approx. Comment. Math. 52 (2015), 299-309. · Zbl 1388.11076
[6] S. Fujima and H. Ichimura: Note on the class number of the \(p\) th cyclotomic field, II, Exp. Math. 27 (2018), 111-118. · Zbl 1427.11113
[7] J.M. Grau, A.M. Oller-Marcén and D. Sadornil: A primarity test for \(Kp^n+1\) numbers, Math. Comp. 84 (2015), 505-512. · Zbl 1352.11105
[8] H. Hasse: Über die Klassenzahl abelscher Zahlkörper, Akademia Verlag, Berlin, 1952. Reprinted with an introduction by J. Martine, Springer, Berlin, 1985.
[9] K. Horie: The ideal class group of the basic \({\Bbb Z}_p\)-extension over an imaginary quadratic field, Tohoku Math. J. 57 (2005), 375-394. · Zbl 1128.11051
[10] H. Ichimura: A note on the relative class number of the cyclotomic \({\Bbb Z}_p\)-extension of \({\Bbb Q}(\sqrt{-p})\), II, Proc. Japan Acad. Ser. A 89 (2013), 21-23. · Zbl 1334.11084
[11] H. Ichimura: Note on Bernoulli numbers associated to some Dirichlet character of prime conductor, Arch. Math. (Basel) 107 (2016), 595-601. Zentralblatt MATH: 1378.11094
Digital Object Identifier: doi:10.1007/s00013-016-0981-4
· Zbl 1378.11094
[12] H. Ichimura: Note on the class number of the \(p\) th cyclotomic field, III, Funct. Approx. Comment. Math. 57 (2017), 93-103. · Zbl 1427.11114
[13] H. Ichimura: Triviality of Iwasawa module associated to some abelian fields of prime conductors, Abh. Math. Semin. Univ. Hambg. 88 (2018), 51-66. Zentralblatt MATH: 1429.11198
Digital Object Identifier: doi:10.1007/s12188-017-0186-1
· Zbl 1429.11198
[14] H. Ichimura and S. Nakajima: A note on the relative class number of the cyclotomic \({\Bbb Z}_p\)-extension of \({\Bbb Q}(\sqrt{-p})\), Proc. Japan Acad. Ser. A 88 (2012), 16-20. · Zbl 1333.11103
[15] S. Jakubec, M. Pasteka and A. Schinzel: Class number of real Abelian fields, J. Number Theory 148 (2015), 365-371. Zentralblatt MATH: 1360.11118
Digital Object Identifier: doi:10.1016/j.jnt.2014.09.027
· Zbl 1360.11118
[16] D.H. Lehmer: Prime factors of cyclotomic class numbers, Math. Comp. 31 (1977), 599-607. Zentralblatt MATH: 0357.12006
Digital Object Identifier: doi:10.1090/S0025-5718-1977-0432589-6
· Zbl 0357.12006
[17] S.R. Louboutin: Lower bounds for relative class numbers of imaginary abelian number fields and CM fields, Acta Arith. 121 (2006), 199-220. Zentralblatt MATH: 1122.11053
Digital Object Identifier: doi:10.4064/aa121-3-1
· Zbl 1122.11053
[18] T. Metsänkylä: Some divisibility results for the cyclotomic class number, Tatra Mt. Math. Publ. 11 (1997), 59-68. · Zbl 0978.11060
[19] T. Metsänkylä: An application of the \(p\)-adic class number formula, Manuscripta Math. 93 (1997), 481-498. · Zbl 0886.11061
[20] O. Ramaré: Approximate formulae for \(L(1,\chi)\), Acta Arith. 100 (2001), 245-266. · Zbl 0985.11037
[21] R. Schoof: Minus class groups of the fields of the \(\ell\) th roots of unity, Math. Comp. 67 (1998), 1225-1245. · Zbl 0902.11043
[22] P. Stevenhagen: Class number parity for the \(p\) th cyclotomic field, Math. Comp. 63 (1994), 773-784. · Zbl 0819.11050
[23] H. Wada and M. Saito: A Table of Ideal Class Groups of Imaginary Quadratic Fields, Sophia Kokyuroku in Mathematics 28, Sophia Univ., Tokyo, 1988. Zentralblatt MATH: 0629.12003
· Zbl 0629.12003
[24] L.C. Washington: The non-\(p\)-part of the class number in a cyclotomic \({\Bbb Z}_p\)-extension, Invent. Math. 49 (1978), 87-97. · Zbl 0403.12007
[25] L.C. Washington: Introduction to Cyclotomic Fields, second edition, Springer, New York, 1997. Zentralblatt MATH: 0966.11047
· Zbl 0966.11047
[26] H.C. Williams and C.R. Zarnke: Some prime numbers of the forms \(2A3^n+1\) and \(2A3^n-1\), Math. Comp. 26 (1972), 995-998. Zentralblatt MATH: 0259.10005
· Zbl 0259.10005
[27] The PARI Group, PARI/GP version 2.
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