数域的tame核的阶及结构

The tame kernel of a number field, that is, the K_2 group of the ring of algebraic integers of a number field is an important and popular research object in algebraic K theory. The research focused on the structure of the tame kernels of number fields before. But the order of the tame kernel of the number field is also important, for example, it is closely related to the famous Lichtenbaum conjecture. This project intends to study the order of the tame kernel of the quadratic field and the structure of the tame kernel of the relative quadratic expansion. On the one hand, compared with the Guass conjecture about the ideal number of the quadratic field, the relation between the order of the tame kernel of the quadratic field and its discriminant is studied. For the real quadratic field, fields with small orders of the tame kernels are given by calculation. For the imaginary quadratic field, it is proved that there are infinite imaginary quadratic fields with trivial p-rank of tame kernel, where p is a prime number. It is proposed to study the relation between the p-rank of the tame kernel of the quadratic field and the p-rank of its ideal group. On the other hand, the Qin’s method is extended to the cyclic quartic field, by the tame symbol theory, and the relation between the 4-rank of its tame kernel and the Legrand symbol matrix is established. It is proposed to study the method of discriminating the 4th order element of the tame kernel of the cyclic quartic field.

数域的tame核，即数域的代数整数环的K_2群是代数K理论中重要且热门的研究对象。以往的研究集中在有关数域的tame核的结构上。但同样重要的是关于数域的tame核的阶数问题，例如其与著名的Lichtenbaum猜想密切相关。本项目拟研究二次域的tame核的阶及相对二次扩张的tame核的结构。一方面，类比Guass关于二次域的理想类数的猜想，研究二次域的tame核的阶与其判别式之间的关系。对于实二次域，通过计算给出具有tame核的阶较小的域。对于虚二次域，证明其tame核的p-rank是平凡的虚二次域有无穷多个，其中p是素数。拟研究二次域的tame核的p-rank与其理想类群的p-rank之间的确定关系。另一方面，将Qin的方法推广到循环四次域上，利用tame符号理论建立其tame核的4-rank与Legrand符号矩阵之间的关系式。拟研究循环四次域的tame核的4阶元的判别方法。

数域的tame核，即数域的代数整数环的K_2群是代数K理论中重要且热门的研究对象。以往的研究集中在有关数域的tame核的结构上。但同样重要的是关于数域的tame核的阶数问题，例如其与著名的Lichtenbaum猜想密切相关。.本项目研究了二次域的tame核的阶及相对二次扩张的tame核的结构。一方面，类比Guass关于二次域的理想类数的猜想，研究了二次域的tame核的阶与其判别式之间的关系。对于实二次域，通过计算给出具有tame核的阶较小的域。对于虚二次域，证明其tame核的p-rank是平凡的虚二次域有无穷多个，其中p是素数。研究了二次域的tame核的p-rank与其理想类群的p-rank之间的确定关系。.另一方面，研究了K2群的相关理论，主要集中在分圆域的整数环的K2群，给出了Herbrand-Ribet定理的K2的类比以及Vandiver猜想的K2的类比。进一步基于环的K2群中比较重要的一类环Dedekind整环的性质研究了矩阵代数的广义逆的表示与结构。.综上所属，我们取得了阶段性的研究成果，并发表了一系列代表性的论文。