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.