On tame and wild kernels of special number fields
The wild kernel \(WK_2(F)\) of a number field \(F\) is the kernel of the map from the \(K\)-group \(K_2(F)\cong F^*\otimes F^*/\langle \{x\otimes (1-x)\}\rangle\) to the sum of all local Hilbert symbols, which properly contains div\((K_2(F))=\bigcap K_2(F)^n\) (the group of divisible elements) when \(F\) is special (otherwise (\(WK_2(F)\;:\) div\((K_2(F))\))=2 ). For special \(F\), the author shows that \(WK_2(F)\subset K_2(F)^{2^b}\) with \(b=\)max\(\{n \mid \zeta _{2^n} +\zeta^{-1}_{2^n}\in F\}\), and \(b\) is the minimum such integer. Then he judges whether an element of \(WK_2(F)\) belongs to certain power of \(K_2(O_F)\) (the tame kernel), determines those \(F\) for which there exist `exotic' Steinberg symbols with values in a finite cyclic group, and indicates how to construct these exotic symbols.
scientific article; zbMATH DE number 2091718