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

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- by Stephen B. Watt
- Trans. Amer. Math. Soc.
**288**(1985), 851-859 - DOI: https://doi.org/10.1090/S0002-9947-1985-0776409-X
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## 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.## References

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## Bibliographic Information

- © Copyright 1985 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**288**(1985), 851-859 - MSC: Primary 11R32; Secondary 11R11
- DOI: https://doi.org/10.1090/S0002-9947-1985-0776409-X
- MathSciNet review: 776409