## Journal of the Mathematical Society of Japan

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

Kuniaki HORIE

#### 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 $\varOmega$ 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 $\varOmega$. 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 $\varOmega$.

#### 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

#### 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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