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Radical Modules over a Dedekind Domain

Published online by Cambridge University Press:  22 January 2016

A. Fröhlich
A. Fröhlich*
Affiliation:
King’s College, London
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A radical of a field K is a non zero element of a given algebraic closure some positive power of which lies in K. The group R(K) of radicals reflects properties of the field K and is in turn easily determined as an extension of the multiplicative group K* of non zero elements of K. The elements of the quotient group R(K)/K* are then conveniently identified with certain subspaces of the algebraic closure, the radical spaces of K (cf. §1). What we are here concerned with is the corresponding arithmetic situation, in which we start with a Dedekind domain o with quotient field K. The role of the radicals is taken over by the radical modules. These form a group (o) which contains the group of fractional ideals of o (cf. §4).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 27 , Issue 2 , July 1966 , pp. 643 - 662
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1966

References

[1] Fröhlich, A., Discriminants of algebraic number fields, Math. Z., 74, 18–28 (1960).Google Scholar
[2] Fröhlich, A., Ideals in an extension field as modules over the algebraic integers in a finite number field, Math Z., 74, 28–38 (1960).CrossRefGoogle Scholar
[3] Fröhlich, A., The module structure of Kummer extensions over Dedekind domains, Journ. f.d. reine u. ang. Math., 209, 39–53 (1962).Google Scholar
[4] Local fields, Proceedings of the Summer school on algebraic number theory, Brighton 1965, to be published by Academic Press.Google Scholar
[5] Hecke, E., Algebraische Zahlen, 1923.Google Scholar
[6] Serre, J-P., Corps locaux, 1962.Google Scholar
[7] Zariski-Samuel, , Commutative algebra, Vol. I, 1958. King’s College, London Google Scholar