  One of the fundamental questions in current field theory, related to
Grothendieck's conjecture of birational anabelian geometry, is the
investigation of the precise relationship between the Galois theory of fields
and the structure of the fields themselves. In this paper we initiate the
classification of additive properties of multiplicative subgroups of fields
containing all squares, using pro-2-Galois groups of nilpotency class at most
2, and of exponent at most 4. This work extends some powerful methods and
techniques from formally real fields to general fields of characteristic not 2.
