  In this paper we completely classify which graded polynomial R-algebras in
finitely many even degree variables can occur as the singular cohomology of a
space with coefficients in R, a 1960 question of N. E. Steenrod, for a
commutative ring R satisfying mild conditions. In the fundamental case R = Z,
our result states that the only polynomial cohomology rings over Z which can
occur, are tensor products of copies of H^*(CP^\infty;Z) = Z[x_2],
H^*(BSU(n);Z) = Z[x_4,x_6,...,x_{2n}], and H^*(BSp(n):Z) =
Z[x_4,x_8,...,x_{4n}] confirming an old conjecture. Our classification extends
Notbohm's solution for R = F_p, p odd. Odd degree generators, excluded above,
only occur if R is an F_2-algebra and in that case the recent classification of
2-compact groups by the authors can be used instead of the present paper. Our
proofs are short and rely on the general theory of p-compact groups, but not on
classification results for these.
