## Osaka Journal of Mathematics

### On some properties of Galois groups of unramified extensions

#### Abstract

Let $k$ be an algebraic number field of finite degree and $k_{\infty}$ be the maximal cyclotomic extension of $k$. Let $\tilde{L}_{k}$ and $L_{k}$ be the maximal unramified Galois extension and the maximal unramified abelian extension of $k_{\infty}$ respectively. We shall give some remarks on the Galois groups $\mathrm{Gal}(\tilde{L}_{k}/k_{\infty})$, $\mathrm{Gal}(L_{k}/k_{\infty})$ and $\mathrm{Gal}(\tilde{L}_{k}/k)$. One of the remarks is concerned with non-solvable quotients of $\mathrm{Gal}(\tilde{L}_{k}/k_{\infty})$ when $k$ is the rationals, which strengthens our previous result.

#### Article information

Source
Osaka J. Math., Volume 53, Number 2 (2016), 321-330.

Dates
First available in Project Euclid: 27 April 2016

https://projecteuclid.org/euclid.ojm/1461781790

Mathematical Reviews number (MathSciNet)
MR3492801

Zentralblatt MATH identifier
1350.11096

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

#### Citation

Asada, Mamoru. On some properties of Galois groups of unramified extensions. Osaka J. Math. 53 (2016), no. 2, 321--330. https://projecteuclid.org/euclid.ojm/1461781790

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