Research On The Tame Kernel Of Number Field Of Degree8

Let E/Q be a Galois extension of number fields with the quater-nion Galois group Q8or Z/2Z×Z/2Z×Z/2Z. In this paper, we prove some relations connecting tame kernels of E with its subfields.In Chapter1, we introduce necessary preliminaries and the de-velopment outline of tame kernel, and give the main results of this paper.Let E/Q be a Galois extension of number fields with the quater-nion Galois group Q8and p an odd prime such that p≡3(mod4). In Chapter2, we investigate the divisibility of p-rank of the tame kernel of E. In particular, if E there is at most one quadratic sub-field such that the p-Sylow subgroup of the tame kernel is nontrivial, then pr-rank (K2OE)-pr-rank (K2OK) is even, where r≥1and K is the unique quartic subfield of E. And we calculate some pr-ranks of K2OE by GP/PARI (p=3,7).Let E/Q be a Galois extension of number fields with the Ga-lois group Z/2Z×Z/2Z×Z/2Z. In Chapter3, we prove the rela-tions between the second regulators R2(·) of E and its subfields. By the Brauer-Kuroda relation of Dedekind zeta function, we get the relationship equation of tame kernels between E and its subfields, i.e. where Q(E) is power of2, k2(E), k2(Fi)(i=0,1,2,3,4,5,6) are the order of K2OE,K2OFi(i=0,1,2,3,4,5,6) respectively, Fi(i0,1,2,3,4,5,6) is the quartic subfield of E.