Restricted ramification for imaginary quadratic number fields and a multiplicator free group

Abstract:

Let $K$ be an imaginary quadratic number field with unit group ${E_K}$ and let $\ell$ be a rational prime such that $\ell \nmid \left | {{E_K}} \right |$. Let $S$ be any finite set of finite primes of $K$ and let $K(\ell ,S)$ denote the maximal $\ell$-extension of $K$ (inside a fixed algebraic closure of $K$) which is nonramified at the finite primes of $K$ outside $S$. We show that the finitely generated pro-$\ell$-group $\Omega (\ell ,S) = \operatorname {Gal}(K(\ell ,S)/K)$ has the property that a complete set of defining relations for $\Omega (\ell ,S)$ as a pro-$\ell$-group can be obtained by lifting the nontrivial abelian or torsion relations in the maximal abelian quotient group $\Omega {(\ell ,S)^{{\text {ab}}}}$. In addition we use the key idea of the proof to derive some interesting results on towers of fields over $K$ with restricted ramification.