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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

11R37 Class field theory
11S31 Class field theory; \(p\)-adic formal groups
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