On the Class Field Theory on Algebraic Number Fields with Infinite Degree

Yukiyosi KAWADA

doi:10.2969/jmsj/00310104

Yukiyosi KAWADA

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1951 Volume 3 Issue 1 Pages 104-115

DOI https://doi.org/10.2969/jmsj/00310104

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Published: 1951 Received: - Available on J-STAGE: August 29, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: August 29, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) We shall call an idèle a∈J(k) a unit idèle if the _??_-componts a(_??_) are units in k(_??_) for all finite prime divisors. Let U(k) be the group of all the unit ideles, then _??_(k)=J(k)/U(k) is isomorphic to the group of all the "umkehrbare Ideale" of k (Cf. Krull [6]).
2) K. Iwasawa, On L-functions, to be published elsewhere, which was read at the meeting of the Math. Soc. Japan, May, 1950.
3) Though J(kλ) are locally compact, J(k) is not always locally compact, and though P(kλ) are discrete subgroups of J(kλ), P(k) is not necessarily discrete. But we can get some properties of J(k) by (22). For example, we can define V*(a) for a∈J(k) by modifying slightly the definition of V(aλ) for aλ∈J(kλ), which was introduced by Artin and Whaples (Cf. E. Artin, W. Whaples, Axiomatic characterization of fields by the product formula for valuations, Bull. Amer. Math. Soc., 51(1945), 469-492. Put J*(k)={a|V*(a)=1}. Then J*(k)⊃P(k), J*(k)⊃ U(k). The structure of the group Jλ(k)/P(k)U(k) is important for the analcgy with the theory of algebraic functions. For these considerations cf. Y. Kawada, On the class field theory on infinite algebraic number fields, (in Japanese). Math. Reports of Tôdai-Kyôyôgakubu, 1 (1950), 85-100.
4) (4.1), (4,2), (4.3), (4,4), (5,1) correspond to Satz 4, Satz 5, Satz 6, Satz 7, and Satz 14 in Moriya [8].

Right : [1] C. Chevalley, Généralisation de la théorie du corps de classes pour les extensions infinies. Journ, de Math., IX, 15 (1936), 357-371.
[2] C. Chevalley, La théorie du corps de classes, Ann. of Math., 41 (1940), 394-418.
[3] H. Freudenthal, Entwicklungen von Räumen und ihre Gruppen, Comp. Math., 4 (1937), 154-234.
[4] J. Herbrand, Théorie arithmetique des corps de nombres de degré infini, I, Math. Ann., 106 (1932), II, 108 (1933).
[5] W. Krull, Galoissche Theorie der unendlichen algebraischen Erweiterungen, Math, Ann., 100 (1928), 687-697.
[6] W. Krull, Idealtheorie in unendlichen algebraischen Zahlkörpern, I, Math. Zeit., 29 (1929), 42-54; II, 31 (1930), 527-557.
[7] M. Moriya, Theorie der algebraischen Zahlkorper unendlichen Grades, Journ. of Fac. Sci. Hokkaido Univ., 3 (1935), 107-140; 4 (1936), 121-122.
[8] M. Moriya, Klassenkörpertheorie im Grossen für unendliche algebraische Zahlkorper Jounr. of Fac. Sci. Hokkaido Univ., 6 (1937), 63-101.
[9] T. Takagi, Über eine Theorie der relativ-Abelschen Zahlkorpers, Journ. Coll. Science, Tokyo, 41 (9), (1920), 1-132.
[10] A. Weil, L'intégration dans les groupes topolcgiques et ses applications, Act. Sci., (1940), Paris.
[11] A. Weil, Sur la théorie du corps de classes, in the same number of this Journ.
[12] C. Whaples, Non analytic class field theory and Grunwald's theorem, Duke Math. Journ., 9 (1942), 455-473.

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