Many questions concerning a zero-dimensional polynomial system can be reduced to linear algebra operations in the quotient algebra A=k[X ₁ ,[^(A)]\widehat{A} . An important feature of our algorithms is that we do not require [^(A)]\widehat{A} to be free and of rank 1. The complexity of our algorithms for computing the minimal polynomial and the rational parametrizations are O(2 ^{nD 5/2} ) and O(n2 ^{nD 5/2} ) respectively, where D is the dimension of A. For fixed n, this is better than algorithms based on linear algebra except when the complexity of the available matrix product has exponent less than 5/2.