| Some Elements of Finite Order in K2Q |
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| 引用本文： | Xiao Yun CHENG Jian Guo XIA Hou Rong QIN. Some Elements of Finite Order in K2Q[J]. 数学学报(英文版), 2007, 23(5): 819-826. DOI: 10.1007/s10114-005-0852-6 |
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| 作者姓名： | Xiao Yun CHENG Jian Guo XIA Hou Rong QIN |
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| 作者单位： | [1]Department of Mathematics, Nanjing University, Nanjing 210093, P. R. China [2]Department of Mathematics, Nanjing Normal University, Nanjing 210097, P. R. China |
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| 基金项目： | This work is supported by SRFDP, the 973 Grant, the National Natural Science Foundation of China 10471118 and the Jiangsu Natural Science Foundation Bk2002023 |
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| 摘 要： | Let K2 be the Milnor functor and let Фn （x）∈ Q[X] be the n-th cyclotomic polynomial. Let Gn（Q） denote a subset consisting of elements of the form {a, Фn（a）}, where a ∈ Q^＊ and {, } denotes the Steinberg symbol in K2Q. J. Browkin proved that Gn（Q） is a subgroup of K2Q if n = 1,2, 3, 4 or 6 and conjectured that Gn（Q） is not a group for any other values of n. This conjecture was confirmed for n =2^T 3S or n = p^r, where p ≥ 5 is a prime number such that h（Q（ζp）） is not divisible by p. In this paper we confirm the conjecture for some n, where n is not of the above forms, more precisely, for n = 15, 21,33, 35, 60 or 105.
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| 关 键 词： | K2Q 有限阶 元素 丢番图方程 |
| 收稿时间： | 2004-11-19 |
| 修稿时间： | 2004-11-192005-03-08 |
Some Elements of Finite Order in K2ℚ |
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| Xiao Yun Cheng,Jian Guo Xia,Hou Rong Qin. Some Elements of Finite Order in K2ℚ[J]. Acta Mathematica Sinica(English Series), 2007, 23(5): 819-826. DOI: 10.1007/s10114-005-0852-6 |
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| Authors: | Xiao Yun Cheng Jian Guo Xia Hou Rong Qin |
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| Affiliation: | (1) Department of Mathematics, Nanjing University, Nanjing 210093, P. R. China;(2) Department of Mathematics, Nanjing Normal University, Nanjing 210097, P. R. China;(3) Department of Mathematics, Nanjing University, Nanjing 210093, P. R. China |
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| Abstract: | Let K 2 be the Milnor functor and let Φ n (x) ∈ ℚ[x] be the n-th cyclotomic polynomial. Let G n (ℚ) denote a subset consisting of elements of the form {a,Φ n (a)}, where a ∈ ℚ*. and {, } denotes the Steinberg symbol in K 2ℚ. J. Browkin proved that G n(ℚ) is a subgroup of K 2ℚ if n = 1, 2, 3, 4 or 6 and conjectured that G n (ℚ) is not a group for any other values of n. This conjecture was confirmed for n = 2 r 3 s or n = p r , where p ≥ 5 is a prime number such that h(ℚ(ζ p )) is not divisible by p. In this paper we confirm the conjecture for some n, where n is not of the above forms, more precisely, for n = 15, 21, 33, 35, 60 or 105. This work is supported by SRFDP, the 973 Grant, the National Natural Science Foundation of China 10471118 and the Jiangsu Natural Science Foundation Bk2002023 |
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| Keywords: | K2Q cyclotomic polynomial Diophantine equation |
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