Content (Tentative)

A quick introduction to notions of rings, fields and other prelims

Euclidean algorithm for GCD of univariate polynomials, Resultant and their properties

Fast Fourier transform, polynomial division, univariate multipoint evaluation

Univariate polynomial factorization over finite fields: CantorZassenhaus and Berlekemp's algorithms

Bivariate polynomial factorization, Hensel lifting and Newton iteration

Multivariate polynomial factorization: Kaltofen's algorithms

Hilbert's irreducibility theorem

Factorization of univariate polynomials over rationals, LLL Basis reduction algorithm

Primality testing: randomized and deterministic algorithms

Multivariate multipoint evaluation over finite fields and a data structure for polynomial evaluation

Ideals, varieties and ideal membership problem

Groebner basis and application to Ideal membership

Emptiness of varieties and weak form of Hilbert's Nullstellansatz

Strong form of Hilbert's Nullstellansatz, Quantifier elimination

Algebraic independence, the Jacobian criterion and testing algebraic independence, applications

Bezout's theorem, Wooley's theorem and applications

Algebraic models of computation, Ben Or and Cleve, Strassen's degree bound, Baur and Strassen

Algebraic P vs NP and connections to its more famous Boolean Cousin

More things if time permits...