### Additive structure of multiplicative subgroups of fields and Galois theory, by Louis Mah'{e}, J'{a}n Min'{a}v{c} and Tara L. Smith

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 $2$.

Louis Mah'{e}, J'{a}n Min'{a}v{c} and Tara L. Smith <Louis.Mahe@univ-rennes1.fr, minac@uwo.ca, tsmith@math.uc.edu>