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 \(h_t^{-}/h_{t-1}^{-}\).
Reviewer: Antonio M. Oller Marcén (Zaragoza)
Software:
PARI/GPReferences:
| [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 · doi:10.13164/ma.2012.01 |
| [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 · doi:10.1090/S0002-9939-97-03909-9 |
| [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 · doi:10.1007/s00013-016-0981-4 |
| [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 · doi:10.1007/s12188-017-0186-1 |
| [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 · doi:10.1016/j.jnt.2014.09.027 |
| [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 · doi:10.1090/S0025-5718-1977-0432589-6 |
| [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 · doi:10.4064/aa121-3-1 |
| [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. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
