I have a couple questions regarding Milnor K-theory.
Given a field $k$ of char $p$, let $k'$ be an Artin-Schreier field extension of $k$. Let's say we know all $K_i^M(k)$, can way recover $K_i^M(k')$? (For example can anything be said about the kernel of the transfer map from $K_i^M(k')$ to $K_i^M(k)$, or maybe is it possible to give a description of generators of $K_i^M(k')$?)
I wonder for function fields of varieties denoted by $K$ (in char $p$) and for an invertible prime $l$, is it true that infinitely $l$ divisible elements in $K_i^M(K)$ have to be torsion? (I wonder a similar question about the etale local rings of varieties at geometric points)