  The quantum nilpotent algebras U^w_-(g), defined by De Concini-Kac-Procesi
and Lusztig, are large classes of iterated skew polynomial rings with rich ring
theoretic structure. In this paper, we prove in an explicit way that all torus
invariant prime ideals of the algebras U^w_-(g) are polynormal. In the special
case of the algebras of quantum matrices, this construction yields explicit
polynormal generating sets consisting of quantum minors for all of their torus
invariant prime ideals. This gives a constructive proof of the Goodearl-Lenagan
polynormality conjecture. Furthermore we prove that Spec U^w_-(g) is normally
separated for all simple Lie algebras g and Weyl group elements w, and deduce
from it that all algebras U^w_-(g) are catenary.
