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On the maximal abelian ℓ-extension of a finite algebraic number field with given ramification

Published online by Cambridge University Press:  22 January 2016

Hiroo Miki
Hiroo Miki*
Affiliation:
Department of Mathematics Faculty of Engineering, Yokohama National University
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Let k be a finite algebraic number field and let ℓ be a fixed odd prime number. In this paper, we shall prove the equivalence of certain rather strong conditions on the following four things (1) ~ (4), respectively :

  1. (1) the class number of the cyclotomic Zℓ-extension of k,

  2. (2) the Galois group of the maximal abelian ℓ-extension of k with given ramification,

  3. (3) the number of independent cyclic extensions of k of degree ℓ, which can be extended to finite cyclic extensions of k of any ℓ-power degree, and

  4. (4) a certain subgroup Bk(m, S) (cf. § 2) of k×/k×)ℓm for any natural number m (see the main theorem in §3).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 70 , July 1978 , pp. 183 - 202
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1978

References

[1] Artin, E. and Tate, J., Class Field Theory, Benjamin, New York, 1967.Google Scholar
[2] Bertrandias, F. and Payan, J.-J., Γ-extensions et invariants cyclotomiques, Ann. scient. ÉC. Norm, Sup., 4e série, 5 (1972), 517–543.Google Scholar
[3] Brumer, A., On the units of algebraic number fields, Mathematika 14 (1967), 121–124.Google Scholar
[4] Brumer, A., Galois groups of extensions of number fields with given ramification, Mich. Math. J. 13 (1966), 33–40.CrossRefGoogle Scholar
[5] Cassels, J. W. S. and Frohlich, A. (Editors), Algebraic number theory, Academic Press, London, 1967.Google Scholar
[6] Gras, M. G., Remarques sur la conjecture de Leopoldt, C. R. Acad. Sc. Paris, t. 274, Série A (1972), 377–380.Google Scholar
[7] Grunwald, W., Ein allgemeines Existenztheorem für algebraische Zahlkörper, J. reine angew. Math. 169 (1933), 103–107.Google Scholar
[8] Hasse, H., Zum Existenzsatz von Grunwald in der Klassenkörpertheorie, J. reine angew. Math. 188 (1950), 40–64.CrossRefGoogle Scholar
[9] Iwasawa, K., A note on the group of units of an algebraic number field, J. Math. pures appl. 35 (1956), 189–192.Google Scholar
[10] Iwasawa, K., A note on class numbers of algebraic number fields, Abh. Math. Sem. Univ. Hamburg 20 (1956), 189–192.Google Scholar
[11] Iwasawa, K., Introduction to Ze-extensions, Lecture notes at Princeton Univ., 1971.Google Scholar
[12] Iwasawa, K., On Ze-extensions of algebraic number fields, Ann. of Math. 98 (1973), 246–326.Google Scholar
[13] Kawada, Y., Class formations, Number theory institute, Amer. Math. Soc. Proc. of Symp. in pure Math. Vol. XX, 1969, 1971, 96–114.Google Scholar
[14] Koch, H., l-Erweiterungen mit vorgegebenen Verzweigungsstellen, J. reine angew. Math. 219 (1965), 30–61.Google Scholar
[15] Koch, H., Galois Theorie der p-Erweiterungen, Springer, Berlin, 1970.Google Scholar
[16] Kubota, T., Galois group of the maximal abelian extension over an algebraic number field, Nagoya Math. J. 12 (1957), 177–189.Google Scholar
[17] Miki, H., On Zp-extensions of complete p-adic power series fields and function fields, J. Fac. Univ. Tokyo Sec. IA, Vol. 21 (1974), 377–393.Google Scholar
[18] Miki, H., A note on the p-rank of ideal class groups of finite algebraic number fields.Google Scholar
[19] Miki, H., On Grunwald-Hasse-Wang’s theorem, J. Math. Soc. Japan 30 (1978), No. 2.Google Scholar
[20] Neukirh, J., Über das Einbettungsproblem der algebraischen Zahlentheorie, Invent. Math. 21 (1973), 59–116.Google Scholar
[21] Roquette, P., On class field towers, [5], Chap. IX, 231–249.Google Scholar
[22] Šafarevič, I. R., Extensions with given points of ramification, Inst. Hautes Études Sci. Publ. Math. 18 (1963), 71–95 = A.M.S. Transl. Ser. 2, 59 (1966), 128–149.Google Scholar
[23] Yokoi, H., On the class number of a relatively cyclic number field, Nagoya Math. J. 29 (1967), 31–44.Google Scholar
[24] Yokoyama, A., On class numbers of finite algebraic number fields, Tôhoku Math. J. 17(1965), 349–357.Google Scholar
[25] Wang, S., On Grunwald’s Theorem, Ann. of Math. 51 (1950), 471–484.Google Scholar
[26] Weil, A., Basic Number Theory, Spring, Berlin, 1967.Google Scholar