Abstract: The Bloch-Kato conjecture is the statement that the Galois cohomology of the absolute Galois group of a field which contains a primitive pth root of unity in mod p coefficients is isomorphic to Milnor K-theory reduced modulo p. This statement is now a theorem proved by Rost and Voevodsky. In particular it says that the cohomology ring of the absolute Galois group is generated by one dimensional classes. It is a natural question to find intermediate Galois extensions of the base field where every element in the cohomology ring decomposes into a sum of products of one dimensional classes. In degree two we answer this question by providing a tower of Galois extensions where indecomposable elements decompose in the next level of the tower. We also illustrate this refinement by directly computing the cohomology rings of superpythagorean fields and p-rigid fields. This is a joint work with J. Minac and S. Chebolu.