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The Notion of Restricted Idèles with Application to Some Extension Fields

Published online by Cambridge University Press:  22 January 2016

Yoshiomi Furuta
Yoshiomi Furuta*
Affiliation:
Mathematical Institute, Kanazawa University
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Let k be an algebraic number field of finite degree, K be its normal extension of degree n, and ŝ be the set of those primes of K which have degree 1. Using this set s instead of the set of all primes of K, we define an s-restricted idèle of K by the same way as ordinary idèles. It is known by Bauer that the normal extension of an algebraic number field is determined by the set of all primes of the ground field which are decomposed completely in the extension field. This suggests that if we treat abelian extensions over K which are normal over k, the class field theory is expressed by means of the ŝ-restricted idèles (theorem 2). When K = k, ŝ is the set of all primes of K, and we have the ordinary class field theory.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 27 , Issue 1 , February 1966 , pp. 121 - 132
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1966

References

[1] Artin, E., Representatives of the connected component of the idèle class group, Proc. Int. Symp. Algebraic Number Theory, Tokyo-Nikko, 1955, 51–54.Google Scholar
[2] Artin, E. and Tate, J., Class field theory, Princeton notes, 1951, distributed by Harvard University.Google Scholar
[3] Bauer, M., Zur Theorie der algebraischen Zahlkörper, Math. Ann., 77 (1916).Google Scholar
[4] Deuring, M., Neuer Beweis des Bauerschen Satzes, J. reine und angew. Math., 173 (1935), 1–4.Google Scholar
[5] Fröhlich, A., The restricted biquadratic residue symbol, Proc. London Math. Soc, (3) 9 (1959), 189–207.Google Scholar
[6] Fröhlich, A., A prime decomposition symbol for certain non Abelian number fields, Acta Sci. Math., 21 (1960), 229–246.Google Scholar
[7] Furuta, Y., A reciprocity law of the power residue symbol, J. Math. Soc. Japan, 10 (1958), 46–54.Google Scholar
[8] Furuta, Y., On meta-abelian fields of a certain type, Nagoya Math. J., 14 (1959), 193–199.CrossRefGoogle Scholar
[9] Furuta, Y., A property of meta-abelian extensions, Nagoya Math. J., 19 (1961), 169–187.Google Scholar
[10] Hasse, H., Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper, II, Jber. Deutsch. Math. Verein., 35 (1926).Google Scholar
[11] Kuroda, S., Über die Zerlegung rationaler Primzahlen in gewissen nicht-abelschen galoischen Körpern, J. Math. Soc. Japan, 3 (1951), 148–156.CrossRefGoogle Scholar
[12] Pontrjagin, L. S., Topological groups. Princeton (1939).Google Scholar
[13] Whaples, G., Non-analytic class field theory and Grunwald’s theorem, Duke Math. J., 9 (1942).Google Scholar