Tame Kernels Of Pure Quintic Number Fields

In this paper,we first study the decomposition of prime ideals of pure quintic fields,and then study the class groups and class numbers of pure quintic fields of Eisenstein type, finally,we study 2-rank and q-rank of the tame kernels of pure quintic fields, where q is an odd prime number.In Chapter 1, we introduce some necessary preliminaries and the background of the development of tame kernels, then we give the main results of this paper.Let, m is a positive integer, then F must be a pure quintic number field. In Chapter 2, we mainly study the decomposition of prime ideals of pure quintic number fields. We also study the ideal class group Cl(OF)and the class number h(F) of pure quintic fields of Eisenstin type.Let q be an odd prime, ?q be a primitive q-th root of unity. Let E?F(?q),then Gal(E/F) = Gal(Q(?q)/Q). In Chapter 3, we mainly study 2-rank and q-rank of the tame kernel of pure quintic number fields and obtain the two results in the following:?:(1) If F is a pure quintic field, i.e., m is a positive integer,then we have: where(2) If F is a pure quintic field, i.e., m is a positive integer,then we have or 2-rank Cl(OF) -1.?:Let F be a pure quintic field, i.e., , m is a positive integer,q an odd prime, E = F(?q) and AE the q-sylow subgroup of Cl(OE). Then we have(1) If q ?5, then we have q-rank K2OF = 9-rank ?q-2AE.(2)If q?5, then we have(a) If , then;(b) If ,then (3) If and, then we have q-rank Especially when which p1,P2, p3, p4 are different primes except , we have obtained the value of 5-rank K2OF.