On The Ring Of Algebraic Integers K <sub> 2 </ Sub> - The Rank Of The Group

Studying the structure of the K₂ -group is one of the basic works of algebraic K -theory. Especially to study the structure of the K₂ -group of O_F of algebraic number field F is a very important work, where O_F is the ringof integers of F.The structure of the cubic number fields is more complicated than that of the quadratic number fields. So it is more difficult to study the structure of tame kernel K₂O_F, of it, where O_F is the ring of integers of the cubic number field F. But the study of it can promote the progress of the study of the structure of K₂O_F of the whole number fields.The main work of this paper is to study the K₂ -group of O_F of the number field F, and discuss two conditions according to whether F contains the primitive root of unity: on the one hand, the pⁿ-rank formula of the K₂O_F of the number field F is given that contains the primitive pⁿth root of unity. On the other hand, the other work of this paper is that we give the information of 19-rank of K₂O_F of cubic cyclic number field F thatdoes not contain the primitive 19th root of unity and only has one ramified prime p.In the first chapter, we give the pⁿ-rank formula of the K₂O_F of the number field F that contains the primitive pⁿth root of unity.The second chapter is the main part of this paper. First, we give some estimates from below and from above of the 19-rank of K₂O_F of cubiccyclic number field F that only has one ramified prime p ( p > 7 ). In many cases, these estimates suffice to determine the structure of the 19-rank.