The Computation Of The TAME Kernel For Global Function Fields

The main work of this dissertation is the study of the tame kernel for function fields with one variable on finite fields (global function fields), namely, the structure of the tame kernel. In fact, this work is bout the structure of the K2 group for global function fields. Since Quillen, Dennis & Stein, and Kahn proved respectively that the K₂ group of discrete valuation rings, Euclidean rings, and ring of integers of global fields are the tame kernel of their quotient fields.From the work of Bass and Tate in 1973, we know that: if the upper bound of the norm of the remaining places is obtained, then after investigating these remaining places by performing some necessary computations with Steinberg symbols, we may get the generators of the tame kernel of the field. Using Bass and Tate's idea, many mathematicians gained some such bounds and successfully computed the tame kernel for a few algebraic number fields.In 2006, the result of Bass and Tate was used in the function field case. Weng proved that for a global function field F with genus g and at least two rational places, the upper bound of the degree (different from but having direct relations with the norm of a place) of the remaining places is 6g-2.Enlightened by Bass, Tate and Weng, we use Chen and You's method for a number field to the function field case and improve Weng's bound. We prove that for a global function field F with at least two rational places the upper bound of the degree of the remaining places is 3 when g=1 and 4g-2 when g>1. At last, we give the explicit generator of the tame kernel for several elliptic function fields.