This monograph presents an intuitive theory of trial wave functions for strongly interacting fermions in fractional quantum Hall states. The correlation functions for the proposed fermion interactions follow a novel algebraic approach that harnesses the classical theory of invariants and semi-invariants of binary forms. This approach can be viewed as a fitting and far-reaching generalization of Laughlin approach to trial wave functions. Aesthetically viewed, it illustrates an attractive symbiosis between the theory of invariants and the theory of correlations. Early research into numerical diagonalization computations for small numbers of electrons shows strong agreement with the constructed trial wave functions. The monograph offers researchers and students of condensed matter physics an accessible discussion of this interesting area of research.