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 OF of algebraic number field F is a very important work, where OF is the ring of integers ofF. 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 K2OF of it, where OF 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 K2OF of the whole number fields. The main work of this paper is to study the K2-group of OF 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 K2OF 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 K2OF of cubic cyclic number field F that does 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 K2OF 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 K2OF of cubic cyclic 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.
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