On local uniformization in arbitrary characteristic, I

We prove that every place of an algebraic function field F|K of arbitrary characteristic admits local uniformization in a finite extension F' of F. We show that F'|F can be chosen to be normal. If K is perfect and P is of rank 1, then alternatively, F' can be obtained from F by at most two Galois extensions; if in addition P is zero-dimensional, then we only need one Galois extension. Certain rational places of rank 1 can be uniformized already on F. We introduce the notion of "relative uniformization" for arbitrary finitely generated extensions of valued fields. Our proofs are based solely on valuation theoretical theorems, which are of fundamental importance in positive characteristic.