Algebraic construction of the tame symbol and the Parshin symbol on a surface

The aim of this work is to offer a new characterization of the tame symbol associated with a discrete valuation from the commutator of a certain central extension of groups. This construction allows us to generalize recent constructions of the tame symbol of an algebraic curve to the arithmetical case, and to offer an easy algebraic proof of a Parshin Reciprocity Law on an algebraic surface with the same method as Tateʹs proof of the Residue Theorem.