This thesis addresses several classic problems in algebraic and symbolic computation related to the solvability of systems of polynomial equations. We develop a parallel algebraic procedure for deciding when a set of multivariate polynomial equations with coefficients in an arbitrary field $K$ have a common solution in an algebraic closure of this field. All computation required by these algorithms takes place over $K$, the field of definition, and hence does not require explicit construction or approximation of solutions. The decision procedure is subsequently extended to yield an algorithm for deciding when solutions exist for arbitrary Boolean combinations of polynomial equations over an algebraically closed field. Modifications are introduced to compute projections of algebraic and semi-algebraic sets, producing an exponential-space algorithm for determining the truth of sentences in the theory of an arbitrary algebraically closed field. In addition, this algorithm can be executed in polynomial space (PSPACE) when restricted to sentences with a bounded number of quantifier alternations. The algebraic nature of the construction also allows us to develop naturally a quantifier elimination procedure for formulas in this theory within similar time and space bounds. Finally, we show that these results are nearly optimal in a common model of parallel arithmetic computation. We also show how these methods can be used to compute the dimension of an arbitrary algebraic set. A variety of other applications-including the construction and approximation of solutions for systems of multivariate polynomial equations and the isolation of real zones-are investigated.