Tohoku Mathematical Journal

Certain primary components of the ideal class group of the $\boldsymbol{Z}_p$-extension over the rationals

Kuniaki Horie

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Abstract

We study, for any prime number $p$, the triviality of certain primary components of the ideal class group of the $\boldsymbol{Z}_p$-extension over the rational field. Among others, we prove that if $p$ is $2$ or $3$ and $l$ is a prime number not congruent to $1$ or $-1$ modulo $2p^2$, then $l$ does not divide the class number of the cyclotomic field of $p^u$th roots of unity for any positive integer $u$.

Article information

Source
Tohoku Math. J. (2), Volume 59, Number 2 (2007), 259-291.

Dates
First available in Project Euclid: 18 June 2007

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1182180736

Digital Object Identifier
doi:10.2748/tmj/1182180736

Mathematical Reviews number (MathSciNet)
MR2347423

Zentralblatt MATH identifier
1202.11050

Subjects
Primary: 11R29: Class numbers, class groups, discriminants
Secondary: 11R18: Cyclotomic extensions 11R20: Other abelian and metabelian extensions 11R23: Iwasawa theory

Keywords
Ideal class group $\boldsymbol{Z}_p$-extension cyclotomic field class number formula decomposition field

Citation

Horie, Kuniaki. Certain primary components of the ideal class group of the $\boldsymbol{Z}_p$-extension over the rationals. Tohoku Math. J. (2) 59 (2007), no. 2, 259--291. doi:10.2748/tmj/1182180736. https://projecteuclid.org/euclid.tmj/1182180736


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References

  • F. J. van der Linden, Class number computations of real abelian number fields, Math. Comp. 39 (1982), 693--707.
  • K. Horie, Ideal class groups of Iwasawa-theoretical abelian extensions over the rational field, J. London Math. Soc. (2) 66 (2002), 257--275.
  • K. Horie, Primary components of the ideal class group of the $\boldsymbolZ_p$-extension over $\boldsymbolQ$ for typical inert primes, Proc. Japan Acad. Ser. A Math. Sci. 81 (2005), 40--43.
  • K. Horie, The ideal class group of the basic $\boldsymbolZ_p$-extension over an imaginary quadratic field, Tohoku Math. J. (2) 57 (2005), 375--394.
  • T. Takagi, Lectures on elementary theory of numbers (in Japanese), Kyoritsushuppansha, Tokyo, 1971.
  • I. M. Vinogradov, Elements of number theory (English translation), Dover Publishing, New York, 1954.
  • L. C. Washington, Class numbers and $\boldsymbolZ_p$-extensions, Math. Ann. 214 (1975), 177--193.