Spaces with polynomial mod-p cohomology

In the early seventies, Steenrod posed the question: which polynomial algeb ras over the Steenrod algebra appear as the cohomology ring of a topologica l space? For odd primes, work of Adams and Wilkerson and Dwyer, Miller and Wilkerson showed that all such algebras are given as the mod-p, reduction o f the invariants of a pseudo reflection group acting on a polynomial algebr a over the p-adic integers. We show that this necessary condition is also s ufficient for finding a realization of such a polynomial algebra. We also s how that, up to completion, every space with polynomial mod-p cohomology is equivalent to the product of the classifying space of a connected compact Lie group and some spaces determined by irreducible p-adic rational pseudo reflection groups.