A henselian valued field K is called a tame field if its separable-algebraic closure K^sep is a tame extension, that is, K^sep is equal to the ramification field of the normal extension K^sep|K. Every algebraically maximal Kaplansky field is a tame field, but not conversely. We prove Ax-Kochen-Ershov Principles for tame fields. This leads to model completeness and completeness results relative to value group and residue field. As the maximal immediate extensions of tame fields will in general not be unique, the proofs have to use much deeper valuation theoretical results than those for other classes of valued fields which have already been shown to satisfy Ax-Kochen-Ershov Principles. We also sketch several applications, including the structure theory of immediate extensions, the Zariski space of places of an algebraic function field, and the model theory of large fields.

Last update: February 4, 1999