Fields with vanishing $K_2$. Torsion in $H^1(X,K_2)$ and $Ch^2(X)$

This paper describes fields $F$ of nonzero characteristic with the property that for all finite extensions $E/F$ $K_2E=0$. We consider a somewhat wider class of fields which includes finite and separably closed fields. For smooth projective varieties $X$ over such a field we show that the groups $H^1(X,K_2)\{l\}$ and $H^2(X_{et},\mathbf Q_l|\mathbf Z_l(2))$, $NH^3(X_{et},\mathbf Q_l|\mathbf Z_l(2))$ and $Ch^2(X)\{l\}$ are isomorphic. These results are applied to describe the groups $SK_1$ of a smooth affine curve over such a field.