数域中tame核的p-秩

The p-rank of tame kernels of number fields

假设E/F是数域的Galois扩张,其Galois群Gal(E/F)为G,p是素数且p#G.利用有限群的特征理论,本文证明了有关tame核K2OE的p-秩的可除性的结果.如果K2OE是循环群,我们得到了#K2OE的上界.特别地,当G=Cn,Dn,A4时,我们利用群的特征表得到了tame核K2OE的p-秩的可除性的结果.假设E/Q是以Dl为Galois群的正规扩张,F/Q是E/Q的l次非正规子扩张,其中l是奇素数.作为一个运用,我们证明了f|p-秩K2OF,其中f是满足pf≡±1(modl)的最小正整数.

Let E/F be a Galois extension of number fields with Galois group G =Gal（E/F）, and let p be a prime not dividing #G. In this paper, using character theory of finite groups, we obtain the upper bound of #K20E if the group K20E is cyclic, and prove some results on the divisibility of the p-rank of the tame kernel K2OE, where E/F is not necessarily abelian. In particular, in the case of G = Cn, Dn, A4, we easily get some results on the divisibility of the p-rank of the tame kernel K20E by the character table. Let E/Q be a normal extension with Galois group Dz, where 1 is an odd As an application, we show that f ｜ p-rank K20F, （rood l）. prime, and F/Q a non-normal subextension with degree l. where f is the smallest positive integer such that p^f= ±1