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On the Ring of Integers in an Algebraic Number Field as a representation Module of Galois Group

Published online by Cambridge University Press:  22 January 2016

Hideo Yokoi
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1. Introduction. It is known that there are only three rationally inequivalent classes of indecomposable integral representations of a cyclic group of prime order l. The representations of these classes are:

  • (I) identical representation,

  • (II) rationally irreducible representation of degree l – 1,

  • (III) indecomposable representation consisting of one identical representation and one rationally irreducible representation of degree l-1 (F. E. Diederichsen [1], I. Reiner [2]).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 16 , February 1960 , pp. 83 - 90
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1960

References

[1] Diederichsen, F. E., Über die Ausreduktion ganzzahliger Gruppendarstellungen bei arithrnetischer Äquivalenz, Abh, Math, Sem. Univ. Hamburg, 14 (1938).Google Scholar
[2] Reiner, I., Integral representations of cyclic groups of prime order, Proc. Amer. Math. Soc., 8 (1957).Google Scholar
[2a] Leopoldt, H. W., Über die Hauptordnung der ganzen Elemente eines abelschen Zahklkörpers, J. Reine Angew, Math., 201 (1959).Google Scholar
[3] Speiser, A., Gruppendeterminante und Körperdiscriminante, Math. Ann., 77 (1916).Google Scholar
[4] Noether, E., Normalbasis bei Körpern ohne höhere Verzweigung, J. Reine Angew. Math., 167 (1932).Google Scholar
[5] Takagi, T., Daisûteki Seisûron, Tokyo (1949), p. 110.Google Scholar
[6] Hasse, H., Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper I a, Jahre. Deutsch. Math. Verein. § 9, s. 75.Google Scholar
[7] Artin, E., Algebraic numbers and algebraic functions I, Princeton (1951), p. 66.Google Scholar
[8] Hasse, H., Zahlentheorie, Berlin (1949), s. 322.Google Scholar
[9] Nakayama, T., On modules of trivial cohomology over a finite group II, Nagoya Math. J., 12 (1957).Google Scholar
[10] Nakayama, T., Cohomology of class field theory and tensor product modules I, Ann. of Math., 65 (1957).Google Scholar
[11] Nakayama, G. Hochschild-T., Cohomology in class field theory, Ann. of Math., 55 (1952).Google Scholar
[12] Leopoldt, H. W., Über Einheitengruppe und Klassenzahl reeller Abelscher Zahlkörper, Abh. Deutsch. Akad. d. Wiss. zu Berlin, Math.-Naturw. Kl., Jahrg. 1953, Nr. 2 (1954).Google Scholar
[13] Chevalley, C., L’arithmétique dans les algebres des Matrices, Actual. Sci. Ind., 323 (1936).Google Scholar