Journal of the Mathematical Society of Japan

Primary components of the ideal class group of an Iwasawa-theoretical abelian number field

Kuniaki HORIE

Full-text: Open access

Abstract

Let S be a non-empty finite set of prime numbers, and let F be an abelian extension over the rational field such that the Galois group of F over some subfield of F with finite degree is topologically isomorphic to the additive group of the direct product of the p -adic integer rings for all p in S . Let m be a positive integer that is neither congruent to 2 modulo 4 nor divisible by any prime number outside S but divisible by all prime numbers in S . Let Ω denote the composite of p n -th cyclotomic fields for all p in S and all positive integers n . In our earlier paper [3], it is shown that there exist only finitely many prime numbers l for which the l -class group of F is nontrivial and the m -th cyclotomic field contains the decomposition field of l in Ω . We shall prove more precise results providing us with an effective upper bound for a prime number l such that the l -class group of F is nontrivial and that the m -th cyclotomic field contains the decomposition field of l in Ω .

Article information

Source
J. Math. Soc. Japan, Volume 59, Number 3 (2007), 811-824.

Dates
First available in Project Euclid: 5 October 2007

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1191591859

Digital Object Identifier
doi:10.2969/jmsj/05930811

Mathematical Reviews number (MathSciNet)
MR2344829

Zentralblatt MATH identifier
1128.11052

Subjects
Primary: 11R29: Class numbers, class groups, discriminants
Secondary: 11R23: Iwasawa theory 11R27: Units and factorization

Keywords
abelian number field ideal class group Iwasawa theory

Citation

HORIE, Kuniaki. Primary components of the ideal class group of an Iwasawa-theoretical abelian number field. J. Math. Soc. Japan 59 (2007), no. 3, 811--824. doi:10.2969/jmsj/05930811. https://projecteuclid.org/euclid.jmsj/1191591859


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References

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