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Normal Bases in Galois Extensions of Number Fields

Published online by Cambridge University Press:  22 January 2016

S. Ullom
S. Ullom*
Affiliation:
University of Maryland, College Park, Maryland
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The notion of module together with many other concepts in abstract algebra we owe to Dedekind [2]. He recognized that the ring of integers OK of a number field was a free Z-module. When the extension K/F is Galois, it is known that K has an algebraic normal basis over F. A fractional ideal of K is a Galois module if and only if it is an ambiguous ideal. Hilbert [4, §§105-112] used the existence of a normal basis for certain rings of integers to develop the theory of root numbers — their decomposition already having been studied by Kummer.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 34 , March 1969 , pp. 153 - 167
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1969

References

[1] Artin, E. and Tate, J., Class field theory, Harvard (1961).Google Scholar
[2] Dirichlet, P.G.L. and Dedekind, R., Vorlesungen über Zahlentheorie, 4th ed., supplement 11, Friedr. Vieweg und Sohn (1894).Google Scholar
[3] Fröhlich, A., The module structure of Kummer extensions over Dedekind domains, J. reine angew. Math. 209 (1962), 39–53.Google Scholar
[4] Hilbert, D., Gesammelte Abhandlungen, vol. 1, Chelsea (1965). (Die Théorie der algebraischen Zahlkörper, Jahresber. der Deutsch. Math. Ver. 4 (1897), 175–546).Google Scholar
[5] Kuroda, S. and Ullom, S., Root numbers associated with normal bases (in preparation).Google Scholar
[6] Leopoldi, H-W., Über die Hauptordnung der ganzen Elemente eines abelschen Zahlkörpers, J. reine angew. Math. 201 (1959), 119–149.Google Scholar
[7] Noether, E., Normalbasis bei Körpern ohne höhere Verzweigung, J. reine angew. Math. 167 (1931), 147–152.Google Scholar
[8] Rim, D.S., Modules over finite groups, Ann. of Math. 69 (1959), 700–712.Google Scholar
[9] Speiser, A., Gruppendeterminante und Körperdiskriminante, Math. Ann. 77 (1916), 546–562.CrossRefGoogle Scholar
[10] Weiss, E., Algebraic number theory, McGraw-Hill (1963).Google Scholar
[11] Yokoi, H., On the ring of integers in an algebraic number field as a representation module of Galois group, Nagoya Math. J. 16 (1960), 83–90.Google Scholar
[12] Yokoi, H., A cohomological investigation of the discriminant of a normal algebraic number field, Nagoya Math. J. 27 (1966), 207–211.Google Scholar