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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Author: Stephen B. Watt
Journal: Trans. Amer. Math. Soc. 288 (1985), 851-859
MSC: Primary 11R32; Secondary 11R11
MathSciNet review: 776409
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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\vert {{E_K}} \right\vert$. 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.

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Article copyright: © Copyright 1985 American Mathematical Society

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