A cohomological translation of the Kaplansky radical for profinite groups

The Kaplansky radical of a field consists of the nonzero elements represented by every norm quadratic form in two variables.

D.~Kijima and M.~Nishi conjectured that, for quadratic extensions, the Kaplansky radicals are related by the norm map in a manner analogous to Hilbert's Theorem~90.

Although this H-conjecture was disproved by K.J.~Becher and D.B.~Leep, it is known to hold for several important classes of fields.

We introduce a cohomological analogue of the Kaplansky radical for arbitrary profinite groups and primes $p$, defined as the orthogonal of $\mathrm{H}^1(G,\mathbb{F}_p)$ with respect to the cup product with itself.

For absolute Galois groups, this recovers the classical Kaplansky radical when $p=2$ and the $p-$radical of Dario--Engler for arbitrary p.

We also formulate a group-theoretic analogue of the H-conjecture, proving that, for fields, it is equivalent to the original conjectural property and depends only on the maximal pro-$2$ quotient of the absolute Galois group.

We establish this property for broad classes of fields, including local and global fields, rational function fields, and all fields whose maximal pro-$p$ Galois group is of elementary type.

Beyond its arithmetic origins, we investigate the property for general pro-$p$ groups, proving its stability under several natural group-theoretic constructions and obtaining new examples, including generalized right-angled Artin pro-$p$ groups and fundamental pro-$p$ groups of suitable graphs of groups, many of which cannot occur as maximal pro-$p$ Galois groups.