Valuations and local uniformization

Abstract

We give the principal notions in valuation theory, the value group and the residue field of a valuation, its rank, the compositions of valuations, and we give some classical examples. Then we introduce the Riemann-Zariski variety of a field, with the topology defined by Zariski. In the last part we recall the result of Zariski on local uniformization and give a sketch of the proof in the case of an algebraic surface.