Galois cohomology of a number field is Koszul

We prove that the Milnor ring of any (one-dimensional) local or global field K modulo a prime number l   is a Koszul algebra over Z/lZ/l. Under mild assumptions that are only needed in the case l=2l=2, we also prove various module Koszulity properties of this algebra. This provides evidence in support of Koszulity conjectures for arbitrary fields that were proposed in our previous papers. The proofs are based on the Class Field Theory and computations with quadratic commutative Gröbner bases (commutative PBW-bases).