A Galois approach to Kaplansky Radical $\times$ Hilbert's Theorem 90

TL;DR

This paper extends Hilbert's Theorem 90 to fields with non-trivial Kaplansky radical by analyzing the Galois group of their maximal 2-extensions as a finitely generated elementary type pro-2 group.

Contribution

It introduces a Galois-theoretic approach to generalize Hilbert's Theorem 90 for fields with non-trivial Kaplansky radical.

Findings

Proves a version of Hilbert's Theorem 90 in this new setting

Characterizes the Galois group as a finitely generated elementary type pro-2 group

Establishes connections between Kaplansky radical and Galois cohomology

Abstract

This paper aims to prove a version of the Hilbert's Theorem 90 for a field with non-trivial Kaplansky radical and the Galois group of its maximal $2$-extension as a finitely generated elementary type pro-2 group.