  Let E: y^2 = x^3 + Ax + B be an elliptic curve defined over a finite field of
characteristic p\geq 3. In this paper we prove that the coefficient at
x^{p(p-1)/2} in the p-th division polynomial \psi_p(x) of E equals the
coefficient at x^{p-1} in (x^3 + Ax + B)^{(p-1)/2}. The first coefficient is
zero if and only if the division polynomial has no roots, which is equivalent
to E being supersingular. Deuring (1941) proved that this supersingularity is
also equivalent to the vanishing of the second coefficient. So the zero loci of
the coefficients (as functions of A and B) are equal; the main result in this
paper is clearly stronger than this last statement.
