On Some Computation Of Tame Kernel

The computation of K2 of the ring of algebraic integers of a number field, i.e. the tame kernel of a number field, is an important topic in the algebraic K-theory. We focus on the computation of the tame kernels of imaginary cyclic. quartic fields with class number one. Moreover, we also expect to take the opportunity to build a software architecture as well as an application program of computing the tame kernels of general fields.It is Tate who proposed the basic method of computing the tame kernels, which was generalized by Hourong Qin, Skalba, Browkin, Belabas and Gangl in different way to some extent. As a result, lots of tame kernels were determined for imaginary quadratic fields and few of imaginary quartic fields, however, with relatively small discriminants and degrees; thus it takes so small amount of calculation that the works can be done by hand or by some easily written programs, while the fields we consider now are much more complicated than those. So. on the one hand we must try our best to improve the theoretical bound, and on the other hand we must also introduce some new ideas about Software Engineering into our computation. Moreover, it is our accommodation of choice to make full use of the high performance of computer hardware; thus we can deal with the large amount of data being processed in computation and more larger-scale calculation so that we can compute the tame kernels for more complicated fields.The first work we have done is to study the tame kernel of quartic field F=Q(?5), and we prove that the tame kernel is trivial, which confirms a conjecture proposed by Browkin under the assumption of Lichtenbaum conjecture. During our study, we make full use of the PARI/GP's advantage in computing some invariants of algebraic number theory so that we can do the work which cannot be completed by hand, and at the same time, we realize fully that more strong ability of computer is highly expected in the computation of tame kernels. This work is covered in Chapter 3.The second work we have finished is to try to build a software architecture for com-puting the tame kernel of a general number field, with the cases of imaginary cyclic quartic fields with class number one as examples. In this study, we realize that as the diserimi-nant and degree of extension of number fields get larger, the amount of computation of tame kernels grows explosively. So, to meet this difficulty, based fully on PARI we intro-duce into our computation the object-oriented method in Software Engineering and the Multi-threaded Parallel Technology in computer science and hence develop a component-based program, which can be used to some extent to compute the tame kernels of any tame kernels and, moreover, which is extensible and reusable and can be extended to an architecture of a program for computing the tame kernels of a general field. Thus the architecture can be regarded as an embryonic form for computing the tame kernel of any number field. Moreover, through deploying the program to a larger computing server, not only we can prove faster that the tame kernel of F= Q(?5) is trivial, but also we can prove in an allowable time that the tame kernels of F=Q(i(?)) (with with discriminant D(F)=2917) and F=Q(i(?)) are both trivial. This work is introduced in Chapter 5.