Hey, maybe I know what you are looking for. Bogomolov and Tschinkel have shown that Milnor K1 (= multiplicative group) AND K2 determine the field - well, assuming its a field of a certain type (see the abstract),

http://www.math.nyu.edu/~tschinke/papers/yuri/09milnor/milnor12.pdf

It's a pretty recent result.

They also touch the question whether maps between these groups come from actual geometric maps between the fields.

Hey, maybe I know what you are looking for. Bogomolov and Tschinkel have shown that (Milnor, doesnt really matter...) K1 (= multiplicative group) AND K2 determine the field - well, assuming its a field of a certain type (see the abstract),

http://www.math.nyu.edu/~tschinke/papers/yuri/09milnor/milnor12.pdf

It's a pretty recent result.

They also touch the question whether maps between these groups come from actual geometric maps between the fields.