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Zbl 0732.11058
Travesa, Artur
Nombre d'extensions abéliennes sur ${\bbfQ}$. (The number of abelian extensions over ${\bbfQ})$. (French)
[J] Sémin. Théor. Nombres Bordx., Sér. II 2, No.2, 413-423 (1990). ISSN 0989-5558

For a finite and non-void set P of prime numbers, the author studies the set $\Sigma\sb{ab}(n,P)$ (resp. $\Sigma\sb{ab}(n,\{e\sb p\},P))$ of abelian number fields of degree n, unramified outside P (resp. having $e\sb p$ as ramification indices for $p\in P)$. He gives necessary and sufficient conditions, in terms of numerical invariants, for $\Sigma\sb{ab}(n,\{e\sb p\},P)$ (resp. $\Sigma\sb{ab}(n,P))$ to be non- void. The idea is to construct cyclotomic extensions L/${\bbfQ}$ belonging to $\Sigma\sb{ab}(\Pi e\sb p,\{e\sb p\},P)$, which are ``universal'' in the sense that any $K\in \Sigma\sb{ab}(n,\{e\sb p\},P)$ is contained in a unique such L. The constructions are explicit enough to give a combinatorial formula for the order of $\Sigma\sb{ab}(n,P).$ \par The obvious generalization would be to replace the base field ${\bbfQ}$ by an arbitrary number field F, and use class-field theory instead of the Kronecker-Weber theorem. However, as is well-known in the inverse Galois problem, serious difficulties would come from the presence of roots of unity in F.
[T.Nguyen Quang Do (Besançon)]

MSC 2000:: *11R20 Other abelian and metabelian extensions
11R18 Cyclotomic extensions

Keywords: abelian fields; ramification; abelian number fields; cyclotomic extensions

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