Towards a refinement of the Bloch-Kato conjecture
arxiv(2024)
Abstract
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 𝔽_p coefficients is generated by elements of degree 1 as an
𝔽_p-algebra. Therefore, for a given Galois extension K/F and an
element α in H^*(Gal(K/F), 𝔽_p) there exits a Galois
extension L/F containing K/F such that the inflation of α in
H^*(Gal(L/F), 𝔽_p) belongs to an 𝔽_p-subalgebra of
H^*(Gal(L/F), 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 α in H^*(Gal(K/F), 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.
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