Some Expansion In The Number Of Non-Abelian Field Tame Core

Let E/F be a Galois extension of number frelds with the Galois group D2n or non-abelian groups of order q3.In this paper,we prove some relations connecting tame kernels of E with its subfields.In Chapter1,we introduce some necessary preliminaries and the development of tame kernels,and give the main results of this paper.Let E2n/E be a Galois extension of number fields with Galois group D2n=＜σ,τ｜σ2n-1=1,τ2=1,τστ-1=σ-1＞.In Chapter2, we get where E2is the fixed field of<σ>,E02n-1is the fixed field of<τ>, E12n-1is the fixed field of<στ>,p is an odd prime.As applications, we give some results about the order of the Sylow p-subgroups when E16/E is a Galois extension of number fields with the Galois group D16.For any odd prime q,the two non-abelian groups of order q3are G1and G2up to isomorphism,where G1=＜g1,g2,g3,｜g1q=g2q=g3q=1,g2g1=g1g2g3,g1g3-g3g1,g2g3=g3g2＞,G2=＜g1,g2｜g1q2=1,g2q=1,g2g1=g11+qg2＞.Let E/F be a non-abelian Galois extension of number fields of degree q3.In Chapter3,We give some expressions for the order of the Sylow p-subgroup of tame kernel of E and some of its subfields containing F.For any prime p,p≠q,we get｜K2(OE)｜q+1｜K2(OF)｜q2the orders of the Sylow p-subgroups of tame kernels when E/Q((?)) is a Galois extension of number fields with non-abelian Galois group of order27.