  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 $p$th root of unity, the Galois cohomology ring of $F$
with $\mathbb{F}_p$ coefficients is generated by elements of degree 1 as an
$\mathbb{F}_p$-algebra. Therefore, for a given Galois extension $K/F$ and an
element $\alpha$ in $H^*(\text{Gal}(K/F), \mathbb{F}_p)$ there exits a Galois
extension $L/F$ containing $K/F$ such that the inflation of $\alpha$ in
$H^*(\text{Gal}(L/F), \mathbb{F}_p)$ belongs to an $\mathbb{F}_p$-subalgebra of
$H^*(\text{Gal}(L/F), \mathbb{F}_p)$ generated by $1$-dimensional classes. It
is interesting to find relatively small explicit Galois extensions $L/F$ with
the above property for a given $\alpha$ in $H^*(\text{Gal}(K/F), \mathbb{F}_p)$
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.
