Let G be the absolute Galois group of a field that contains a primitive p-th root of unity. This is a profinite group which is a central 
 object of study in arithmetic algebraic geometry. In joint work with Ido Efrat and Jan Minac, we have shown that a remarkably small 
 quotient of this big group determines the entire Galois cohomology of G. As application of this result, we give new examples of 
 profinite groups that are not realisable as absolute Galois groups of fields. I will present an overview of this work.