Free Pro-2 Product Decompositions and Hilbert's Theorem 90 for the Kaplansky Radical

Abstract

The H-conjecture of Kijima and Nishi predicts a Hilbert Theorem~90 type property for the Kaplansky radical with respect to quadratic field extensions. In this paper, we establish a Galois-theoretic criterion ensuring this property for fields F with finitely many square classes. The criterion is formulated in terms of a free pro-2 product decomposition of the maximal pro-2 Galois group G_F(2). As an application, we prove the H-conjecture for fields whose maximal pro-2 Galois groups are of elementary type, in the sense of the pro-2 version of Efrat's Elementary Type Conjecture. Our criterion also applies to pre-2-henselian fields, to the previously known positive cases established by Iwakami, Kijima, and Nishi, and to the families of fields with nontrivial Kaplansky radical constructed by Berman and Kula. It thereby provides a unified Galois-theoretic explanation for the currently known positive instances of the finite-dimensional H-conjecture.