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On the deduction of the class field theory from the general reciprocity of power residues

Published online by Cambridge University Press:  22 January 2016

Tomio Kubota  and
Satomi Oka
Tomio Kubota
Affiliation:
Department of Mathematics, Meijo University, Shiogamaguchi 1-501, Tenpaku-ku Nagoya, 468-8502, Japan
Satomi Oka
Affiliation:
Department of Mathematics, Meijo University, Shiogamaguchi 1-501, Tenpaku-ku, Nagoya, 468-8502, Japan
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Abstract

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We denote by (A) Artin’s reciprocity law for a general abelian extension of a finite degree over an algebraic number field of a finite degree, and denote two special cases of (A) as follows: by (AC) the assertion (A) where K/F is a cyclotomic extension; by (AK) the assertion (A) where K/F is a Kummer extension. We will show that (A) is derived from (AC) and (AK) only by routine, elementarily algebraic arguments provided that n = (K : F) is odd. If n is even, then some more advanced tools like Proposition 2 are necessary. This proposition is a consequence of Hasse’s norm theorem for a quadratic extension of an algebraic number field, but weaker than the latter.

Keywords

Primary: 11R37Secondary: 11E12
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 160 , 2000 , pp. 135 - 142
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2000

References

[1] Chevalley, C., Deux théorèmes d’arithmétique, J. Math. Soc., Japan, 3–1 (1951), 36–44.Google Scholar
[2] Kubota, T., Geometry of numbers and class field theory, Japan. J. Math., 13–2 (1987), 235–275.Google Scholar