Homogeneous skew-fields of non-commutative rational functions and their reduced Whitehead groups

A construction of skew-fields of non-commutative rational functions is studied. We discuss and prove criterions for such skew-fields to be homogeneous and finite-dimensional over their centers and describe relations between some objects defined in terms of the skew-fields of constants, which help to compute reduced Whitehead groups of corresponding skew-fields of non-commutative rational functions. In particular we present a proof of one previous result of V. P. Platonov and the author about reduced Whitehead groups of skew-fields of non-commutative rational functions announced in 1979 and obtain in non-Henselian case of such skew-fields analogues of all results of Yu. L. Ershov for Henselian situation.