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K_2 group of torsion fields

Author: LiuMin
Tutor: XuKeJian
School: Qingdao University
Course: Basic mathematics
Keywords: K2 group cyclotomic element Browkin’s conjecture ABC theorem for function fields
CLC: O154.3
Type: Master's thesis
Year: 2008
Downloads: 11
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Abstract


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 {ξn,a}（a∈F*） in K2（F）, and proved that if F is a global field containing the n-th primitive root of unityξn, then every element of order n in K2（F） can be written in the form of {ξn,a}（a∈F*）. Suslin generalized Tate’s result to any field containingξn. To generalize the result to a field possibly withoutξn, Browkin considered the elements of the form {a,Φn（a）} in K2（F） , called cyclotomic elements, whereΦn（X） denotes the n-th cyclotomic polynomial and proved that if n=1, 2, 3, 4 or 6 and F≠F2, then Gn（F） is a subgroup of K2（F）. Then, Browkin proposed that for any integer n≠1, 2, 3, 4, 6 and any field F, Gn（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 Gln（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 unityξn; 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 Gln （F） is not a subgroup of K2（F） for a function field F.

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CLC: > Mathematical sciences and chemical > Mathematics > Algebra,number theory, portfolio theory > Category theory, homological algebra > Algebraic K- theory
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