Domain K <sub> 2 </ Sub> Group Of Torsion

It is important to determine the torsion elements of the K₂ group of a field in the K-theory. J. Tate studied the elements of the form {ξ_n,a}(a∈F^*) in K₂(F), and proved that if F is a global field containing the n-th primitive root of unityξ_n, then every element of order n in K₂(F) can be written in the form of {ξ_n,a}(a∈F^*). Suslin generalized Tate's result to any field containingξ_n. To generalize the result to a field possibly withoutξ_n, Browkin considered the elements of the form {a,Φ_n(a)} in K₂(F) , called cyclotomic elements, whereΦ_n(X) denotes the n-th cyclotomic polynomial and proved that if n=1, 2, 3, 4 or 6 and F≠F₂, then G_n(F) is a subgroup of K₂(F). Then, Browkin proposed that for any integer n≠1, 2, 3, 4, 6 and any field F, G_n(F) is not a subgroup of K₂(F), in particular, G₅(Q)is not a subgroup of K₂(Q). That is the Browkin's conjecture.In this paper, we partially prove the Browkin's conjecture for function fields over non- algebraically-closed fields. Firstly, we construct infinitely many different elements in K₂(F). Secondly, we reduce the Browkin's conjecture to a problem about rational points on finitely many curves. And then, by the ABC theorem for function fields, we prove that G_lⁿ(F) is not a subgroup of K₂(F), where l is a prime and n≥3.In chapter 1, we introduce Tate's results about the elements of order n in K₂(F), where F is a global field containing the n-th primitive root of unityξ_n; in chapter 2, we recall the Browkin's conjecture and the current progress, particularly the results obtained by Xu Kejian; in chapter 3, using the ABC theorem we prove that G_lⁿ (F) is not a subgroup of K₂(F) for a function field F.