Transfer maps and nonexistence of joint determinant

TL;DR

This paper demonstrates the equivalence of transfer maps in Milnor's K-theory and Goodwillie groups, leading to an isomorphism that helps describe joint determinants of commuting matrices over various fields.

Contribution

It establishes the functorial equivalence of transfer maps in Milnor's K-theory and Goodwillie groups, enabling a new understanding of joint determinants for matrices over different fields.

Findings

Transfer maps in Milnor's K-theory and Goodwillie groups are equivalent.

An explicit isomorphism between Goodwillie groups and Milnor's K-groups is constructed.

Joint determinants for matrices over finite, rational, real, and complex fields are explicitly described.

Abstract

Transfer Maps, sometimes called norm maps, for Milnor's $K$-theory were first defined by Bass and Tate (1972) for simple extensions of fields via tame symbol and Weil's reciprocity law, but their functoriality had not been settled until Kato (1980). 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, the field of rational numbers, real numbers and complex numbers into the respective group of units of given field.