Nagoya Mathematical Journal

On the rank of the first radical layer of a $p$-class group of an algebraic number field

Hiroshi Yamashita

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Let $p$ be a prime number. Let $M$ be a finite Galois extension of a finite algebraic number field $k$. Suppose that $M$ contains a primitive $p$th root of unity and that the $p$-Sylow subgroup of the Galois group $G=Gal(M/k)$ is normal. Let $K$ be the intermediate field corresponding to the $p$-Sylow subgroup. Let ${\frak g}=Gal(K/k)$. The $p$-class group ${\cal C}$ of $M$ is a module over the group ring ${\bf Z}_p G$, where ${\bf Z}_p$ is the ring of $p$-adic integers. Let $J$ be the Jacobson radical of ${\bf Z}_p G$. ${\cal C}/J {\cal C}$ is a module over a semisimple artinian ring ${\bf F}_p {\frak g}$. We study multiplicity of an irreducible representation $\Phi$ apperaring in ${\cal C}/J{\cal C}$ and prove a formula giving this multiplicity partially. As application to this formula, we study a cyclotomic field $M$ such that the minus part of ${\cal C}$ is cyclic as a ${\bf Z}_p G$-module and a CM-field $M$ such that the plus part of ${\cal C}$ vanishes for odd $p$.

To show the formula, we apply theory of central extensions of algebraic number field andstudy global and local Kummer duality between the genus group and the Kummer radical for the genus field with respect to $M/K$.

Article information

Nagoya Math. J., Volume 156 (1999), 85-108.

First available in Project Euclid: 27 April 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11R37: Class field theory
Secondary: 11R29: Class numbers, class groups, discriminants 11R34: Galois cohomology [See also 12Gxx, 19A31]


Yamashita, Hiroshi. On the rank of the first radical layer of a $p$-class group of an algebraic number field. Nagoya Math. J. 156 (1999), 85--108.

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  • D.J. Benson, Representations and cohomology I, Cambrige studies in advanced math. 30, Cambrige (1991).
  • C.W. Curtis and I. Reiner, Methods of representation theory with applications to finite groups and orders, Vol. I, John Wiley & Sons, Inc. (1981).
  • R. Greenberg, On the Iwasawa invariants of totally real number fields , Amer. J. Math., 98 (1976), 263–284.
  • W. Jehne, On knots in algebraic number theory , J. reine angew. Math., 311/312 (1979), 215–254.
  • G. Karpilovsky, Group representations, Vol I, Part A and Part B, North-Holland math. studies 175 (1992, North-Holland).
  • H.W.Leopoldt, Zur Struktur der $l$-Klassengruppe galoissher Zahlkörper , J. reine angew. Math., 199 (1958), 165–174.
  • K. Miyake, Central extensions and Schur's multiplicators of Galois groups , Nagoya Math. J., 90 (1983), 137–144.
  • S. Shirai, On the central class field $\rm mod\ \frak m$ of Galois extensions of algebraic number field , Nagoya Math. J., 71 (1978), 61–85.
  • W. Sinnott, On the Stickelberger ideal and the circular units of a cyclotomic field , Ann. of Math., 108 (1978), 107–134.
  • H. Yamashita, On the Iwasawa invariants of totally real number fields , manuscripta math., 79 (1993), 1–5.