On tame and wild kernels of special number fields

Special number fields are those number fields F for which the wild kernel properly contains the group of divisible elements of K2(F). We examine the relationship between the wild kernel, the tame kernel and the group of divisible elements for such number fields. For a special number field F we show that WK2(F) K2(F)2b, but WK2(F) K2(F)2b+2 where b=b+(F)=max{n ζ2n+ζ2n−1 F}. We examine analogous questions for instead of K2(F). As an application, we determine those number fields for which there exist ‘exotic’ Steinberg symbols with values in a finite cyclic group and we show how to construct these exotic symbols.