## Tokyo Journal of Mathematics

### On the Structure of the Galois Group of the Maximal Pro-$p$ Extension with Restricted Ramification over the Cyclotomic $\mathbb{Z}_p$-extension

Tsuyoshi ITOH

#### Advance publication

This article is in its final form and can be cited using the date of online publication and the DOI.

#### Abstract

Let $k_\infty$ be the cyclotomic $\mathbb{Z}_p$-extension of an algebraic number field $k$. We denote by $S$ a finite set of prime numbers which does not contain $p$, and $S(k_\infty)$ the set of primes of $k_\infty$ lying above $S$. In the present paper, we will study the structure of the Galois group $\mathcal{X}_S (k_\infty)$ of the maximal pro-$p$ extension unramified outside $S (k_\infty)$ over $k_\infty$. We mainly consider the question whether $\mathcal{X}_S (k_\infty)$ is a non-abelian free pro-$p$ group or not. In the former part, we treat the case when $k$ is an imaginary quadratic field and $S = \emptyset$ (here $p$ is an odd prime number which does not split in $k$). In the latter part, we treat the case when $k$ is a totally real field and $S \neq \emptyset$.

#### Article information

Source
Tokyo J. Math., Advance publication (2019), 24 pages.

Dates
First available in Project Euclid: 24 August 2019

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1566612101

Subjects
Primary: 11R23: Iwasawa theory

#### Citation

ITOH, Tsuyoshi. On the Structure of the Galois Group of the Maximal Pro-$p$ Extension with Restricted Ramification over the Cyclotomic $\mathbb{Z}_p$-extension. Tokyo J. Math., advance publication, 24 August 2019. https://projecteuclid.org/euclid.tjm/1566612101

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