Transfer maps and nonexistence of joint determinant

Transfer Maps, sometimes called norm maps, for Milnor’s K-theory were first defined by Bass and Tate [H. Bass, J. Tate, The Milnor ring of a global field, in: Algebraic K-theory, II: “Classical” algebraic K-theory and connections with arithmetic, Proc. Conf., Seattle, Wash., Battelle Memorial Inst., 1972, Lecture Notes in Math., vol. 342, Springer, Berlin, 1973, pp. 349–446] for simple extensions of fields via tame symbol and Weil’s reciprocity law, but their functoriality had not been settled until Kato [Kazuya Kato, A generalization of local class field theory by using K-groups. II, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (3) (1980) 603–683]. On the other hand, functorial transfer maps for the Goodwillie group are easily defined. We show that these natural transfer maps actually agree with the classical but difficult transfer maps by Bass and Tate. With this result, we build an isomorphism from the Goodwillie groups to Milnor’s K-groups of fields, which in turn provides a description of joint determinants for the commuting invertible matrices. In particular, we explicitly determine certain joint determinants for the commuting invertible matrices over a finite field, Q, R and C into the respective group of units of given field.