On the structure of the Galois group of the maximal pro-$p$ extension with restricted ramification over the cyclotomic $\mathbb{Z}_p$-extension
| Autor: | Tsuyoshi Itoh |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Mathematics - Number Theory
Group (mathematics) General Mathematics Mathematics::Number Theory 010102 general mathematics Galois group Prime number Structure (category theory) Extension (predicate logic) Algebraic number field 01 natural sciences Combinatorics 11R23 0103 physical sciences FOS: Mathematics Quadratic field 010307 mathematical physics Number Theory (math.NT) 0101 mathematics Finite set Mathematics |
| Zdroj: | Tokyo J. Math. 43, no. 1 (2020), 181-204 |
| DOI: | 10.48550/arxiv.1810.10268 |
| Popis: | 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 $S \neq \emptyset$. Comment: 20 pages, changed several places, added sentences and references |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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