We prove that every place of an algebraic function field of arbitrary characteristic admits a local uniformization in a finite extension of the function field. We give a valuation theoretical description of these extensions; in certain cases, they can be found in the henselization of the function field. For places satisfying the Abhyankar equality and for discrete rational places, no extension is needed if the base field is perfect. Our proof is based solely on valuation theoretical theorems which are of fundamental importance in positive characteristic. It provides additional assertions (which in general do not hold without extensions).

Last update: February 3, 1999