The concept of a resolvent of a prime ideal was originally introduced by J. F. Ritt along with the notion of a characteristic set. The motivation for studying resolvents comes from its connections with the birational isomorphisms that describe structures of irreducible algebraic varieties by means of an equivalent hypersurface and a one-to-one rational map. As a result, these ideas have a wide range of applications in such areas as solid modeling, computer-aided design and manufacturing. An algorithm to compute the resolvent by means of characteristic sets was first proposed by Ritt. This and some related algorithms have resurfaced as interest in resolvent structures have grown, spurred by its applicability. Unfortunately, the algebraic complexity of the resolvent and the computational complexity of the associated algorithms have never been explicitly explored. In this paper, we give single exponential upper and lower bounds for the degrees of the resolvent and its associated parametrizing polynomials. We also show that the resolvent can be computed deterministically in single exponential sequential and polynomial parallel time complexity. All previous algorithms for resolvent had relied on a random choice of certain extraneous parameters.