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Sur la formule du produit pour le symbole de reste normique généralisé. (On the product formula for the generalized norm residue symbol). (French) Zbl 0584.12006
Let L/K be a finite extension of number fields. For each place $$\nu$$ of K, let $$L_{\nu}$$ denote the intersection (in some algebraic closure of $$K_{\nu})$$ of the completions of L over $$\nu$$, and let $$L_{\nu}^{ab}$$ be the maximal sub-extension of $$L_{\nu}$$ abelian over $$K_{\nu}$$. Let $$D_{\nu}^{ab}(L/K)$$ be the Galois group of the latter. For each place $$\nu$$, local class field theory gives a map of $$K^{\times}$$ to $$D_{\nu}^{ab}(L/K)$$. Letting $$\nu$$ run over all places of K gives a map from $$K^{\times}$$ to $$\oplus_{\nu}D_{\nu}^{ab}(L/K)$$, such that the product of the components of an element corresponding to a given x in K is equal to 1 when restricted to the maximal abelian extension $$L^{ab}$$ of K.
The point of this paper is to prove the converse of the last fact. Namely, the only families $$(\sigma_{\nu})$$ of $$\oplus_{\nu}D_{\nu}^{ab}(L/K)$$ whose restrictions to $$L^{ab}$$ satisfy the product formula, $$\prod_{\nu}\sigma_{\nu}|_{L^{ab}}=1$$, are those arising from a single element x of $$K^{\times}$$ by the procedure above.
Reviewer: L.G.Roberts

##### MSC:
 11R37 Class field theory 11S31 Class field theory; $$p$$-adic formal groups
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