This fact actually follows from de Jong's result. Our proofs are based solely on valuation theoretical theorems, which are of fundamental importance in positive characteristic. We also indicate certain analogues of our results for the arithmetic case. The abstract is available as dvi file or as postscript file . The model theory of tame fields has nice applications to the theory of places of algebraic function fields in arbitrary characteristic.
All tame fields are perfect.
Similar problems appear for fields of mixed characteristic which are not finitely ramified, like certain infinite algebraic extensions of the p-adics. Relative QE in various forms is known for all above mentioned classes of valued fields, except for the tame fields in general. The idea is to show QE relative to a structure naturally associated to every valued fields, which carries more information than value group and residue field. The class of all henselian fields of mixed characteristic admits QE relative to these structures, because they take care of the essential positive characteristic part of the fields. In special cases, one would like to get away with much less complex reducts of the amc-structures.