The Galois Group Of The Maximal 2-extension Of A Field

This work is on the structure of the Galois groups of the maximal 2-extensions of a field. This work is closely related to many famous open questions (Galois representations, the elementary conjecture, Fermat prime numbers, the level and u-invariant of a field, etc.).;Let F be a field of characteristic not 2 and {dollar}F\sb{lcub}q{rcub}{dollar} the maximal 2-extension of F. Let {dollar}G\sb{lcub}q{rcub}{dollar} be the Galois group of {dollar}F\sb{lcub}q{rcub}/F.{dollar} This work deals with the connections among the Galois theory of F, the structure of {dollar}G\sb{lcub}q{rcub}{dollar} and the mod 2 Galois cohomology of {dollar}G\sb{lcub}q{rcub}.{dollar};One of the most famous questions about Galois cohomology theory and quadratic form theory is Milnor\u27s conjecture which made a fascinating connection among the Milnor K-theory, the mod 2 Galois cohomology and quadratic forms. This conjecture was solved very recently by Voevodsky. We are interested in what properties of {dollar}G\sb{lcub}q{rcub}{dollar} one can deduce from Milnor\u27s conjecture. We are also interested in the Galois theoretic meaning of Milnor\u27s conjecture.;In this work, we find a Galois theoretic meaning of Milnor\u27s conjecture on degree 2. We also obtain a necessary and sufficient condition for {dollar}G\sb{lcub}q{rcub}{dollar} to satisfy the injectivity part of the Milnor conjecture on degree 3 under the assumption of the Milnor conjecture on degree 2.;We also develop new tools for solving inverse Galois problems and solve some of them