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E(K/k) and other arithmetical invariants for finite Galois extensions

Published online by Cambridge University Press:  22 January 2016

Shin-Ichi Katayama
Shin-Ichi Katayama*
Affiliation:
Department of Mathematics, College of General Education, Tokushima University, Tokushima, 770, Japan
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Let k be an algebraic number field and K be a finite extension of k. Recently, T. Ono defined positive rational numbers E(K/k) and E′(K/k) for K/k. In [7], he investigated some relations between E(K/k) and other cohomological invariants for K/k. He obtained a formula when K is a normal extension of k. In our paper [3], we obtained a similar formula for E′(K/k) in the case of normal extensions K/k. Both proofs essentially use Ono’s results on the Tamagawa number of algebraic tori, on which the formulae themselves do not depend. Hence, in [8], T. Ono posed a problem to give direct proofs of these formulae.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 114 , June 1989 , pp. 135 - 142
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1989

References

[1] Casseles, J. and Fröhlich, A., Algebraic Number Theory, Acad. Press, New York, 1967.Google Scholar
[2] Furuta, Y., The genus field and genus number in algebraic number fields, Nagoya Math. J., 29 (1967), 281–285.Google Scholar
[3] Katayama, S., Class number relations of algebraic tori I, Proc. Japan Acad., 62 (1986), 216–218.Google Scholar
[4] Katayama, S., The Euler number and other arithmetical invariants for finite Galois extensions of algebraic number fields, ibid., 63A (1987), 27–30.Google Scholar
[5] Ono, T., Arithmetic of algebraic tori, Ann. of Math., 74 (1961), 101–139.Google Scholar
[6] Ono, T., On the Tamagawa number of algebraic tori, ibid., 78 (1963), 47–73.Google Scholar
[7] Ono, T., On some class number relations for Galois extensions Nagoya Math. J., 107 (1987), 121–133.Google Scholar
[8] Ono, T., Algebraic groups and number theory, Sûgaku, 38 (1986), 218–231 (in Japanese).Google Scholar
[9] Sasaki, R., Some remarks to Ono’s theorem on the generalization of Gauss’s genus theory (preprint).Google Scholar
[10] Weil, A., Adeles and Algebraic Groups, notes by Demazure, M. and Ono, T., Progress in Math., 23, Birkhauser, 1982.Google Scholar
[11] Yokoi, H., On some class number of relatively cyclic number fields, Nagoya Math. J., 29 (1967), 31–44.Google Scholar