Every place admits local uniformization in a finite extension of the function field

We prove that every place P of an algebraic function field F vertical bar K of arbitrary characteristic admits local uniformization in a finite extension T of F. We show that F vertical bar F can be chosen to be Galois, after a finite purely inseparable extension of the ground field K. Instead of being Galois, the extension can also be chosen such that the induced extension, FP vertical bar FP of the residue fields is purely inseparable and the value group of F only gets divided by the residue characteristic. If F lies in the completion of an Abhyankar place, then no extension of F is needed. Our proofs are based solely on valuation theoretical theorems, which are of particular importance in positive characteristic. They are also applicable when working over a subring R subset of K and yield similar results if R is regular and of dimension smaller than 3.