A practical method is given that can determine whether or not it is possible for all or part of a problem in crystal elasticity to be expressed in explicit rational or radical terms when the characteristic sextic polynomial is known to be unsolvable. It involves the determination of the group to which an elastic quantity (displacement, stress, strain, energy density) belongs under permutations of the roots of the sextic polynomial. If this is not the symmetric group then a rational or radical expression is impossible. It is applied to the three-dimensional problem of a point force (the Green's function) in a cubic crystal. It is proved that it is impossible that the Green's function could be given by any radical expression that is valid for all elastic constants and directions in the crystal, as the existence of such an expression would lead to the solution of a proven unsolvable polynomial. It follows that the rational expression given by Dikici (1986) for the Green's function is not just incorrect, but that it is impossible that there could be any such rational expression. The same impossibility is found to apply to the displacements, stresses and strains of a straight dislocation in a cubic crystal. The application to other elasticity problems is discussed.