Let $G$ be the absolute Galois group of a field that contains a primitive $p$th root of unity. This is a very mysterious profinite group which is a central object of study in Galois theory. 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 surprising 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.