Algebra and randomness come together rather nicely in computation. A central example of this relationship in action is the Schwartz-Zippel lemma and its application to the fast randomized checking of polynomial identities. In this thesis, we further this relationship in two ways: (1) by compiling new algebraic techniques that are of potential computational interest, and (2) demonstrating the relevance of these techniques by making progress on several questions in randomness and pseudorandomness. The technical ingredients we introduce include: " Multiplicity-enhanced versions of the Schwartz-Zippel lenina and the "polynomial method", extending their applicability to "higher-degree" polynomials. " Conditions for polynomials to have an unusually small number of roots. " Conditions for polynomials to have an unusually structured set of roots, e.g., containing a large linear space. Our applications include: * Explicit constructions of randomness extractors with logarithmic seed and vanishing "entropy loss". " Limit laws for first-order logic augmented with the parity quantifier on random graphs (extending the classical 0-1 law). " Explicit dispersers for affine sources of imperfect randomness with sublinear entropy.