On Browkin’s conjecture about the elements of order five in <Emphasis Type="Italic">K</Emphasis><Subscript>2</Subscript>(ℚ)

Based on the work of Lenstra, a succinct proof of Browkin’s conjecture about the elements of order five in K ₂(ℚ) is given. This work was supported by the National Natural Science Foundation of China (Grant No. 10371061).     

Neither G9（Q） nor G11（Q） Is a Subgroup of K2（Q）

It is proved that neither G9(Q) nor G11(Q) is a subgroup of K2(Q) which confirms two special cases of a conjecture proposed by Browkin, J. (Lecture Notes in Math., 966, Springer-Verlag, New York, Heidelberg, Berlin, 1982, 1-6).     

Some Elements of Finite Order in K2Q

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.     

某些K_2O_F的结构(英文)

本文研究二次数域 Ｆ＝ 的 Ｋ２ＯＦ结构,其中 ｄ－３ ｍｏｄ ９和 ｄ ≠－３．找到了关于Ｆ＝的Ｋ２ＯＦ的３阶元和Ｆ＝的Ｋ２ＯＦ的生成元,推广Ｔ Ｂａｓｓ和 Ｔａｔｅ的一个定理和给出了 Ｆ＝ 的 Ｋ２ＯＦ的结构以及 ＳＬｎ（ＯＦ）（ｎ≥３）的表示关系．     

A conjecture on a class of elements of finite order in K2Fp

For a local field F the finite subgroups of K₂ F are expressed by a class of special elements of finite order, which makes a famous theorem built by Moore, Carroll, Tate and Merkurjev more explicit and also disproves a conjecture posed by Browkin.     

On K 2-group of a formal matrix ring

Let S be a formal matrix ring, T the subring consisting of all diagonal elements, I the set consisting of all off-diagonal elements. Then I is a split radical ideal under certain conditions. In this paper, we show that K2 (S)■K2 (T) ? K2 (S,I), and a presentation of K2 (S,I) is given.     

Z[(-5)1/2]上不可分的Hermite型

本文讨论了Z[(-5)^1/2]上不可分的正定Hermite型的构作.给出了所有秩为2判别式等于2的不可分的正定Hermite型.当秩n≥3时,证明了存在Z[(-5)^1/2]上判别式等于2的不可分的正定Hermite型,并给出了它们的明显结构.     

Q(61/2)上的定幺模格的分类

利用推广的邻格方法,完成了Q(6^1/2)上秩4的在一个阿基米德除子上正定,在另一阿基米德除子上负定的所有幺模格种的分类.     

Q(6~(1/2))上的定幺模格的分类

利用推广的邻格方法,完成了Q(6~(1/2))上秩4的在一个阿基米德除子上正定,在另一阿基米德除子上负定的所有幺模格种的分类.     

On the typical rank of real binary forms

We determine the rank of a general real binary form of degree d?=?4 or d?=?5. In the case d?=?5, the possible values of the rank of such general forms are 3, 4, and 5. This is the first reported case, to our knowledge, where more than two typical ranks have been found. We prove that a real binary form of degree d with d real roots has rank?d.     

On a conjecture about orders of products of elements in the symmetric group

For any $a,b,c,n\in \mathbb{N}$ with $1<a,b,c\le n?2$, then there exist $\alpha ,\beta \in S_n$ such that $o(\alpha )=a$, $o(\beta )=b$ and $o(\alpha \beta )=c$. This is a conjecture of Stefan Kohl and which is closely related to problem on covers of the complex projective line. In this note we prove the conjecture is true.     

On Jackson's theorem in the space ℓ2(ℤ 2 n )

Estimates of Jackson's constants in the space ℓ₂(ℤ ₂ ⁿ ) are given for the case of approximation by sums of subspaces on which irreducible representations of the isometry group of (ℤ ₂ ⁿ ) act and for the case in which the modulus of continuity is defined using generalized translations. Coding theory results on efficiency estimates for binaryd-codes with respect to the Hamming distance are used. Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 390–405, September, 1996.     

Classification of 9-dimensional trilinear alternating forms over GF(2)

Let V be a finite-dimensional vector space over a finite field and let f be a trilinear alternating form over V. For such forms, we introduce two new invariants. Together with a generalized radical polynomial used for classification of forms in dimension 8 over GF(2), they are sufficient to distinguish between all trilinear alternating forms in dimension 9 over GF(2). To prove the completeness of the list of forms, we computed their groups of automorphisms. There are 31 degenerate and 317 nondegenerate forms. We point out some forms with either small or large automorphism group.     

Classification of 8-Dimensional Trilinear Alternating Forms over GF(2)

Let V be an n-dimensional vector space over a finite field and let f be a trilinear alternating form over V. For such forms we introduce a new invariant called radical polynomial and investigate its behaviour, in particular in the case of the 2-element field. We show that it is compatible with direct products of forms and how it is related to its values on dimension n ? 1. Moreover, it turns out that it is full up to dimension 7. On the other hand, on higher dimensions it is no more full and it is necessary to generalize it to obtain (using computer) a classification of forms on dimension 8 over the 2-element field. This classification is provided, together with the sizes of stabilizers of the corresponding forms.     

The order of elements in Sylow p-subgroups of the symmetric group

Define a random variable ξ_n by choosing a conjugacy class C of the Sylow p-subgroup of S_pn by random, and let ξ_n be the logarithm of the order of an element in C. We show that ξ_n has bounded variance and mean order log n /log p +O(1), which differs greatly from the average order of elements chosen with equal probability. This revised version was published online in June 2006 with corrections to the Cover Date.     

函数域的K_2群的挠元

设F是域,令G_n(F)={{a,φ_n(a)}∈K_2(F)| a,Φ_n(a)∈F~*},这里Φ_n(x)是n次分圆多项式.使用函数域的ABC定理证明了若F是常数域为k函数域,l≠ch(k)是素数,则对n≥3且l>2或n>3且l=2,G_(ln)(F)不是K_2(F)的子群.由此部分地证实了Browkin的猜想.     

关于Q(4k~(2n)+1)的理想类群的循环子群

设ｄ,ａ,ｋ,ｎ是适合４ｋ２ｎ＋１＝ｄａ２,ｋ＞１,ｎ＞２,ｄ无平方因子的正整数;又设Ｃ（Ｋ）和ｈ（Ｋ）分别是实二次域Ｋ的理想类群和类数．本文证明了：当ａ＜０．５ｋ０．５６ｎ时,则ｈ（ｋ）＝０（ｍｏｄｎ）和Ｃ（Ｋ）必有ｎ阶循环子群．