Towards a refinement of the Bloch-Kato conjecture

Rost and Voevodsky proved the Bloch-Kato conjecture relating Milnor k-theory and Galois cohomology. Their result implies that for a field F containing a primitive pth root of unity, the Galois cohomology ring of F with Fₚ coefficients is generated by elements of degree 1 as an Fₚ-algebra. Therefore, for a given Galois extension K/F and an element in H^* (Gal (K/F), Fₚ) there exits a Galois extension L/F containing K/F such that the inflation of in H^* (Gal (L/F), Fₚ) belongs to an Fₚ-subalgebra of H^* (Gal (L/F), Fₚ) generated by 1-dimensional classes. It is interesting to find relatively small explicit Galois extensions L/F with the above property for a given in H^* (Gal (K/F), Fₚ) as above. In this paper, we provide some answers to this question for cohomology classes in degree two, thus setting the first step toward refining the Bloch-Kato conjecture. We illustrate this refinement by explicitly computing the cohomology rings of superpythagorean and p-rigid fields. Additionally, as a byproduct of our work, we characterize elementary abelian 2-groups as the only finite p-groups whose mod-p cohomology ring is generated by degree-one elements. This provides additional motivation for studying refinements of the Bloch-Kato conjecture and exploring the connections between group cohomology and Galois theory.