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 . 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.