For an algebraic number fieldK containing roots of unity of orderp m (p a prime), we study a new kind of reciprocity law, called “primitive”, which allows us, when it exists, to express wild Hilbert symbols of orderp m in terms of a (uniformly) finite set of tame symbols. A sufficient condition for existence is the nullity of thep-group of classes ofp-ideals ofK. Applications are given to the description of the Galois group of the maximal prop-extension ofK which is unramified outside some finite set of primes containingp.
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