The infinitude of the reduced Whitehead group in the Tannaka–Artin problem

Using the methods and results of preceding papers of the author (V. P. Platonov, The Tarmaka–Artin problem and groups of projective conorms, Dokl. Akad. Nauk SSSR, 222, № 6 (1975), 1229–1302; The Tarmaka–Artin problem and reduced $K$-theory, Izv. Akad. Nauk SSSR, Ser. Mat., 40, № 2 (1976), 227–261), in the first part of this paper we find conditions under which the reduced Whitehead group is infinite, and in the second, larger part we give the solution of the Tannaka–Artin problem for cyclic algebras. In particular, we completely calculate the reduced Whitehead group $SK_1(A)$ for cyclic algebras $A$ over formal power series fields and construct cyclic algebras of arbitrary degree $n^2$ with Whitehead group that is arbitrarily large but finite, and also with infinite Whitehead group.
Bibliography: 15 titles.