  We introduce a new method of constructing complete sequences of key
polynomials for simple extensions of tame fields. In our approach the key
polynomials are taken to be the minimal polynomials over the base field of
suitably constructed elements in its algebraic closure, with the extensions
generated by them forming an increasing chain. In the case of algebraic
extensions, we generalize the results to countably generated infinite tame
extensions over henselian but not necessarily tame fields. In the case of
transcendental extensions, we demonstrate the central role that is played by
the implicit constant fields, which reveals the tight connection with the
algebraic case.
