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A generalization of Hubert’s theorem 94

Published online by Cambridge University Press:  22 January 2016

Hiroshi Suzuki
Hiroshi Suzuki*
Affiliation:
Department of Mathematics, Faculty of Science, Tokyo Metropolitan University, Fukasawa Setagaya-ku, Tokyo 158, Japan
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In this paper we shall prove the following theorem conjectured by Miyake in [3] (see also Jaulent [2]).

THEOREM. Let k be a finite algebraic number field and K be an unramified abelian extension of k, then all ideals belonging to at least [K: k] ideal classes of k become principal in K.

Since the capitulation homomorphism is equivalently translated to a group-transfer of the galois group (see Miyake [3]), it is enough to prove the following group-theoretical verison:

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 121 , March 1991 , pp. 161 - 169
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1991

References

[1] Artin, E. and Tate, J., Class Field Theory, Benjamin, 1967.Google Scholar
[2] Jaulent, J.-F., L’état actuel du problem de la capitulation, Séminaire de Théorie des Nombres de Bordeaux, 1987–1988 Exp. no. 17, 1988.Google Scholar
[3] Miyake, K., Algebraic investigations of Hilbert’s theorem 94, the principal ideal theorem and the capitulation problem, Expo. Math., 7 (1989), 289–346.Google Scholar