  The Poincare duality of classical cohomology and the extension of this
duality to quantum cohomology endows these rings with the structure of a
Frobenius algebra. Any such algebra possesses a canonical ``characteristic
element;'' in the classical case this is the Euler class, and in the quantum
case this is a deformation of the classical Euler class which we call the
``quantum Euler class.'' We prove that the characteristic element of a
Frobenius algebra A is a unit if and only if A is semisimple, and then apply
this result to the cases of the quantum cohomology of the finite complex
Grassmannians, and to the quantum cohomology of hypersurfaces. In addition we
show that, in the case of the Grassmannians, the [quantum] Euler class equals,
as [quantum] cohomology element and up to sign, the determinant of the Hessian
of the [quantum] Landau-Ginzbug potential.
