On the Stickelberger ideal and the relative class number

Authors:
Tatsuo Kimura and Kuniaki Horie

Journal:
Trans. Amer. Math. Soc. **302** (1987), 727-739

MSC:
Primary 11R18; Secondary 11R29

MathSciNet review:
891643

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be any imaginary abelian field, the integral group ring of , and the Stickelberger ideal of . Roughly speaking, the relative class number of is expressed as the index of in a certain ideal of described by means of and the complex conjugation of , with a rational number in , which can be described without and is of lower than if the conductor of is sufficiently large (cf. [**6, 9, 10**]; see also [**5**]). We shall prove that , a natural number, divides . In particular, if varies through a sequence of imaginary abelian fields of degrees bounded, then takes only a finite number of values. On the other hand, it will be shown that can take any value in when ranges over all imaginary abelian fields. In this connection, we shall also make a simple remark on the divisibility for the relative class number of cyclotomic fields.

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1987-0891643-8

Keywords:
(Imaginary) abelian field,
Stickelberger ideal,
relative class number,
analytic class number formula

Article copyright:
© Copyright 1987
American Mathematical Society