Cohomology of varieties over the maximal Kummer extension of a number field

Let $X$ be a smooth projective geometrically connected variety defined over a number field $K$. We prove that the geometric étale cohomology of $X$ with $\mathbb{Q}\mathrm{/}\mathbb{Z}$-coefficients has finitely many classes invariant under the Galois group of the maximal Kummer extension of $K$ in odd degrees. In particular, every abelian variety has finite torsion over the maximal Kummer extension. This improves results by Rössler and the second author as well as Murotani and Ozeki. We also show that finiteness of torsion of a given abelian variety over non-abelian solvable extensions of $K$ is not controlled by the Galois group of the extension.