  The structure of Poisson polynomial algebras of the type obtained as
semiclassical limits of quantized coordinate rings is investigated. Sufficient
conditions for a rational Poisson action of a torus on such an algebra to leave
only finitely many Poisson prime ideals invariant are obtained. Combined with
previous work of the first-named author, this establishes the Poisson
Dixmier-Moeglin equivalence for large classes of Poisson polynomial rings, such
as semiclassical limits of quantum matrices, quantum symplectic and euclidean
spaces, quantum symmetric and antisymmetric matrices. For a similarly large
class of Poisson polynomial rings, it is proved that the quotient field of the
algebra (respectively, of any Poisson prime factor ring) is a rational function
field $F(x_1,...,x_n)$ over the base field (respectively, over an extension
field of the base field) with $\{x_i,x_j\}= \lambda_{ij} x_ix_j$ for suitable
scalars $\lambda_{ij}$, thus establishing a quadratic Poisson version of the
Gel'fand-Kirillov problem. Finally, partial solutions to the isomorphism
problem for Poisson fields of the type just mentioned are obtained.
