## Osaka Journal of Mathematics

### On tamely ramified Iwasawa modules for the cyclotomic $\mathbb {Z}_{p}$-extension of abelian fields

Tsuyoshi Itoh

#### Abstract

Let $p$ be an odd prime, and $k_{\infty}$ the cyclotomic $\mathbb{Z}_{p}$-extension of an abelian field $k$. For a finite set $S$ of rational primes which does not include $p$, we will consider the maximal $S$-ramified abelian pro-$p$ extension $M_{S} (k_{\infty})$ over $k_{\infty}$. We shall give a formula of the $\mathbb{Z}_{p}$-rank of $\mathrm{Gal}(M_{S} (k_{\infty})/k_{\infty})$. In the proof of this formula, we also show that $M_{\{q\}} (k_{\infty})/L(k_{\infty})$ is a finite extension for every real abelian field $k$ and every rational prime $q$ distinct from $p$, where $L(k_{\infty})$ is the maximal unramified abelian pro-$p$ extension over $k_{\infty}$.

#### Article information

Source
Osaka J. Math., Volume 51, Number 2 (2014), 513-537.

Dates
First available in Project Euclid: 8 April 2014

https://projecteuclid.org/euclid.ojm/1396966260

Mathematical Reviews number (MathSciNet)
MR3192553

Zentralblatt MATH identifier
1300.11111

Subjects
Primary: 11R23: Iwasawa theory
Secondary: 11R18: Cyclotomic extensions

#### Citation

Itoh, Tsuyoshi. On tamely ramified Iwasawa modules for the cyclotomic $\mathbb {Z}_{p}$-extension of abelian fields. Osaka J. Math. 51 (2014), no. 2, 513--537. https://projecteuclid.org/euclid.ojm/1396966260

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