  In this paper we identify conditions under which the cohomology $H^*(\Omega
M\xi;\k)$ for the loop space $\Omega M\xi$ of the Thom space $M\xi$ of a
spherical fibration $\xi\downarrow B$ can be a polynomial ring. We use the
Eilenberg-Moore spectral sequence which has a particularly simple form when the
Euler class $e(\xi)\in H^n(B;\k)$ vanishes, or equivalently when an orientation
class for the Thom space has trivial square. As a consequence of our
homological calculations we are able to show that the suspension spectrum
$\Sigma^\infty\Omega M\xi$ has a local splitting replacing the James splitting
of $\Sigma\Omega M\xi$ when $M\xi$ is a suspension.
