Louis Mah\'e, J\'an Min\'a\v c,  and Tara L. Smith 
Additive structure of  multiplicative subgroups of fields and Galois theory

\abstract
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 $2$.
\endabstract