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A Generalization of Global Class Field Theory

Published online by Cambridge University Press:  20 November 2018

Tae Kun Seo  and
G. Whaples
Tae Kun Seo
Affiliation:
Indiana University, Bloomington, Indiana
G. Whaples
Affiliation:
University of Kentucky, Lexington, Kentucky University of Massachusetts, Boston, Massachusetts
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Let R be a field of rational functions of one variable over a field of constants R0. Dock Sang Rim (6) has proved that the global reciprocity law in exactly the usual sense holds whenever R0 is an absolutely algebraic quasi-fini te field of characteristic not equal to 0: this was known before only when R0 was a finite field. We shall give another proof of Rim's result by means of a noteworthy generalization of the usual global reciprocity law. Namely, let R0 be a finite field and let F be the set of all fields k contained in some fixed Ralg.clos. and of finite degree over R. The reciprocity law states that there exists a family {fk}, k ∈ F, of functions fk: Ck → G(kabel.clos./k) (where Ck is the idèle class group of k) enjoying certain properties such as the norm transfer law.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 609 - 614
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Artin, E., Algebraic numbers and algebraic functions. I, Mimeographed lecture notes, Princeton University, Princeton, N.J. ; New York University, New York, 1951 (printed éd., Gordon and Breach, New York, 1967).Google Scholar
2. Artin, E. and Tate, J., Class field theory, Lecture notes, Harvard University, Cambridge, Massachusetts, 1961 (W. A. Benjamin, New York, 1968).Google Scholar
3. Kawada, Y., On the class field theory on algebraic number fields with infinite degree, J. Math. Soc. Japan 8 (1951), 104–115.Google Scholar
4. Kawada, Y., Class formations. IV (infinite extension of ground field), J. Math. Soc. Japan 9 (1957), 395–405.Google Scholar
5. Mori, M., Ueber Klas s enkôrper théorie fur unendliche Erweiterungen von einem p-adischen Zahlkôrper, Proc. Japan Acad. 33 (1957), 376–379.Google Scholar
6. Rim, D. S. and Whaples, G., A remark on quasi-finite field, Nagoya Math. J. 27 (1966), 323–329.Google Scholar
7. Kun Seo, Tae, A generalization of global class field theory, Ph.D. Thesis, Indiana University, Bloomington, Indiana, 1965.Google Scholar
8. Whaples, G., Lecture notes on class field theory, Bowdoin College, Summer, 1966.Google Scholar