The fundamental theorem of algebra determines the number of characteristic roots of an ordinary differential equation of integer order. Thismay cease to be true for a differential equation of fractional order.Theresults given in this paper suggest that the number of the characteristic roots of a class of oscillators of fractional order may in general be infinitely great. Further, we infer that it may also be the case for the characteristic roots of a differential equation of fractional order greater than 1.The relationship between the range of the fractional order and the locations of characteristic roots of oscillators in the complex plane is considered.