Journal of the Mathematical Society of Japan
- J. Math. Soc. Japan
- Volume 59, Number 3 (2007), 811-824.
Primary components of the ideal class group of an Iwasawa-theoretical abelian number field
Let be a non-empty finite set of prime numbers, and let be an abelian extension over the rational field such that the Galois group of over some subfield of with finite degree is topologically isomorphic to the additive group of the direct product of the -adic integer rings for all in . Let be a positive integer that is neither congruent to modulo nor divisible by any prime number outside but divisible by all prime numbers in . Let denote the composite of -th cyclotomic fields for all in and all positive integers . In our earlier paper , it is shown that there exist only finitely many prime numbers for which the -class group of is nontrivial and the -th cyclotomic field contains the decomposition field of in . We shall prove more precise results providing us with an effective upper bound for a prime number such that the -class group of is nontrivial and that the -th cyclotomic field contains the decomposition field of in .
J. Math. Soc. Japan, Volume 59, Number 3 (2007), 811-824.
First available in Project Euclid: 5 October 2007
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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