Metsänkylä, Tauno An application of the \(p\)-adic class number formula. (English) Zbl 0886.11061 Manuscr. Math. 93, No. 4, 481-498 (1997). The author proves some results about \(p\)-divisibility of \(h(K)\), the class number of a real abelian field \(K\), using Leopoldt’s class number formula in terms of \(p\)-adic \(L\)-functions. He generalizes a theorem of S. Jakubec [Abh. Math. Semin. Univ. Hamb. 63, 67-86 (1993; Zbl 0788.11052)] to \[ \prod_\chi\sum_j b_j\chi(j)\equiv 0\pmod p \] if \(p|h(K)\) (where \(\chi\neq 1\) runs over Dirichlet characters of \(K\)), by giving a variant of Leopoldt’s results from 1959, that is \(\prod_\chi B_\chi^{(p-1)}\equiv 0\pmod p\) if \(p|h(K)\), with \(B_\chi^{(p-1)}\) the \((p-1)\)st generalized Bernoulli number; and then he shows that \(p\nmid h(K)\) for \(K=\mathbb{Q}(\zeta_q)^+\) under certain conditions. Reviewer: Zhang Xianke (Beijing) Cited in 2 ReviewsCited in 13 Documents MSC: 11R29 Class numbers, class groups, discriminants 11S31 Class field theory; \(p\)-adic formal groups 11R42 Zeta functions and \(L\)-functions of number fields 11S40 Zeta functions and \(L\)-functions 11R18 Cyclotomic extensions 11R20 Other abelian and metabelian extensions Keywords:class number; real abelian field; \(p\)-adic \(L\)-function; \(p\)-divisibility Citations:Zbl 0788.11052 PDFBibTeX XMLCite \textit{T. Metsänkylä}, Manuscr. Math. 93, No. 4, 481--498 (1997; Zbl 0886.11061) Full Text: DOI EuDML References: [1] Apostol, T.: Generalized Dedekind sums and transformation formulae of certain Lambert series. Duke Math. J.17, no. 2, 147–157 (1950) · Zbl 0039.03801 · doi:10.1215/S0012-7094-50-01716-9 [2] Coates, J.:p-adicL-functions and Iwasawa’s theory. Pp. 269–353 in Algebraic number fields (ed. by A. Fröhlich), London: Academic Press 1977 [3] Frobenius, G.: Über die Bernoullischen Zahlen und die Eulerschen Polynome. Sitzungsber. Königl. Preuss. Akad. Wiss. Berlin, 809–847 (1910); pp. 440–478 in Gesammelte Abhandlungen III, Berlin-Heidelberg-New York: Springer 1968. · Zbl 1264.11013 [4] Jakubec, S.: On divisibility of class number of real abelian fields of prime conductor. Abh. Math. Sem. Univ. Hamburg63, 67–86 (1993) · Zbl 0788.11052 · doi:10.1007/BF02941333 [5] Jakubec, S.: On the divisibility ofh + by the prime 3. Rocky Mountain J. Math.24, no. 4, 1467–1473 (1994) · Zbl 0821.11053 · doi:10.1216/rmjm/1181072349 [6] Jakubec, S.: On divisibility ofh + by the prime 5. Math. Slovaca44, no. 5, 651–661 (1994) · Zbl 0827.11071 [7] Jakubec, S.: Connection between the Wieferich congruence and divisibility ofh +. Acta Arith.71, no. 1, 55–64 (1995) · Zbl 0833.11052 [8] Jakubec, S.: Connection between congruencesn q 1 (modq 2) and divisibility ofh +. Abh. Math. Sem. Univ. Hamburg66, 151–158 (1996) · Zbl 0871.11075 · doi:10.1007/BF02940801 [9] Jakubec, S.: On divisibility of the class numberh + of the real cyclotomic fields of prime degreel. Math. Comp. (to appear) · Zbl 0914.11057 [10] Jakubec, S. and Louboutin, S., personal communication [11] Jakubec, S. and Trojovský, P.: On divisibility of the class numberh + of the real cyclotomic fields \(\mathbb{Q}(\zeta _p + \zeta _p ^{ - 1} )\) by primesq<5000. (Manuscript) · Zbl 0895.11044 [12] Leopoldt, H.-W.: Über Klassenzahlprimteiler reeller abelscher Zahlkörper als Primteiler verallgemeinerter Bernoullischer Zahlen. Abh. Math. Sem. Univ. Hamburg23, 36–47 (1959) · Zbl 0086.03103 · doi:10.1007/BF02941024 [13] Leopoldt, H.-W.: Über Fermatquotienten von Kreiseinheiten und Klassenzahlformeln modulop. Rend. Circ. Mat. Palermo (2)9, 39–50 (1960) · Zbl 0098.03501 · doi:10.1007/BF02843700 [14] Leopoldt, H.-W.: Zur Arithmetik in abelschen Zahlkörpern. J. Reine Angew. Math.209, 54–71 (1962) · Zbl 0204.07101 · doi:10.1515/crll.1962.209.54 [15] Leopoldt, H.-W.: Einep-adische Theorie der Zetawerte, II. Diep-adische{\(\Gamma\)}-Transformation. J. Reine Angew. Math.274/275, 224–239 (1975) · Zbl 0309.12009 · doi:10.1515/crll.1975.274-275.224 [16] Slavutski, I. Sh.:L-functions and the class number of cyclotomic fields (Russian). Uspekhi Mat. Nauk43, no. 5 (263), 215–216 (1988) · Zbl 0663.12009 [17] Washington, L. C.: Introduction to cyclotomic fields, 2nd ed. (Graduate texts in mathematics, 83) New York-Berlin-Heidelberg: Springer 1996 · Zbl 0484.12001 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.