Let J be the Jacobian variety of a hyperelliptic curve over $mathbb{Q}$. Let M be the field generated by all square roots of rational integers over a finite number field K. Then we prove that the Mordell-Weil group J(M) is the direct sum of a finite torsion group and a free $mathbb{Z}$-module of infinite rank. In particular, J(M) is not a divisible group. On the other hand, if $widetilde{M}$ is an extension of M which contains all the torsion points of J over $widetilde{mathbb{Q}}$, then $J(widetilde{M}^{sol})/J(widetilde{M}^{sol})_{tors}$ is a divisible group of infinite rank, where $widetilde{M}^{sol}$ is the maximal solvable extension of $widetilde{M}$.