Algebraic Geometry is a huge area of mathematics which went through several phases: Hilbert's fundamental paper from 1890, sheaves and cohomology introduced by Serre in the 1950s, Grothendieck's theory of schemes in the 1960s and so on. This book covers the basic material known before Serre's introduction of sheaves to the subject with an emphasis on computational methods. In particular, we will use Gröbner basis systematically. The highlights are the Nullstellensatz, Gröbner basis, Hilbert's syzygy theorem and the Hilbert function, Bézout’s theorem,  semi-continuity of the fiber...