循环四次域的tame核的Sylow 2-子群

代数整数环的K2群，即数域F的tame 核，是代数K理论中重要而活跃的研究领域。以往秦厚荣等人对二次域的tame核做了大量的工作，对其结构，与类群的关系以及其p-rank等方面有丰富而深入的研究。目前，对于循环四次域的tame核的研究还不是很充分。本项目拟研究循环四次域的tame核的Sylow 2-子群的结构。首先，类比二次域的情况，研究tame核的2-rank与域的判别式素因子个数的关系，进而判定循环四次域的tame核的2阶元的生成元。其次，对tame核的2阶元给出是否为tame核的元素的平方的判定条件，从而得到循环四次域的tame核的Sylow 2-子群是初等Able群的结构定理。最后，讨论循环四次域的tame核的4-rank。

The K2 group of the algebraic integer of a number field , i.e., the tame kernel of number field F, is an important and active field of algebraic K theory. In the past, Qin and other people had done a lot of work on the tame kernels of quadratic fields. They did rich and deeper results, on its structure, its relation with class groups and its p-rank. Now, the study on the tame kernels of cyclic quartic fields is not enough. In this project, we are going to study the Sylow 2-groups of tame kernels of cyclic quartic fields. Firstly, similar to the quadratic fields, we will study the 2-rank of tame kernel and its relation with the number of prime divisors in the discriminant, and then get the generators of tame kernels of order 2. Secondly, for the elements of order 2 in the tame kernel, we will give a method to determine when they are square in the tame kernel, and get a theorem on deciding when the Sylow 2-groups of tame kernels of cyclic quartic fields are elementary Abelian groups. Finally, we will study the 4-rank of the tame kernel.