Let F be a cubic cyclic field with t(2)ramified primes.For a finite abelian group G,let r3(G)be the 3-rank of G.If 3 does not ramify in F,then it is proved that t-1 r3(K2O F)2t.Furthermore,if t is fixed,for any s satisfying t-1 s 2t-1,there is always a cubic cyclic field F with exactly t ramified primes such that r3(K2O F)=s.It is also proved that the densities for 3-ranks of tame kernels of cyclic cubic number fields satisfy a Cohen-Lenstra type formula d∞,r=3-r2∞k=1(1-3-k)r k=1(1-3-k)2.This suggests that the Cohen-Lenstra conjecture for ideal class groups can be extended to the tame kernels of cyclic cubic number fields.
Let F=Q(p^(1/2)),where p=8t+1 is a prime.In this paper,we prove that a special case of Qin's conjecture on the possible structure of the 2-primary part of K_2O_F up to 8-rank is a consequence of a conjecture of Cohen and Lagarias on the existence of governing fields.We also characterize the16-rank of K_2O_F,which is either 0 or 1,in terms of a certain equation between 2-adic Hilbert symbols being satisfied or not.
