Let $F$ be a field that has a primitive $p$-th root of unity. According to the Bloch–Kato conjecture, now a theorem by Voevodsky and Rost, the norm-residue map \begin{equation} k_*(F)/pk_*(F) \rightarrow H^*(F, \mathbb{F}_p) \end{equation} from the reduced Milnor $K$-theory to the Galois cohomology of $F$ is an isomorphism of $\mathbb{F}_p$-algebras.

This isomorphism gives a presentation of the rather mysterious Galois cohomology ring through generators and relations. In joint work with Jan Minac, Cihan Okay, Andy Schultz, and Charlotte Ure, we have obtained a second cohomology refinement of the Bloch–Kato conjecture. Using this we can characterize the maximal $p$-extension of $F$, as the “decomposing field” for the cohomology of the absolute Galois group.