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Let J be the Jacobian variety of a hyperelliptic curve over 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 Z-module of infinite rank. In particular, J(M ) is not a divisible group. On the other hand, if M is an extension of M which contains all the torsion points of J over Q, then J( M sol )/J( M sol )tors is a divisible group of infinitedoi:10.5666/kmj.2009.49.3.419 fatcat:mckvksx53vd5bowkwvgrfhf3rm