On Grobner Bases and Their Applications

This thesis focuses on presenting three different and interesting applications of Grobner basis to commutative algebra, algebraic geometry, and field theory. In particular, we will begin by introducing the fundamental theory of Grobner bases followed by providing an algorithm, widely known as Buchberger's Algorithm, to compute a Grobner basis for an ideal in a polynomial ring followed by applying these concepts to 1. determine a criterion for the membership of a polynomial in the radical of an ideal in a polynomial ring,2. devise an algorithm to obtain a Grobner basis for the kernel and image of a polynomial map, and3. determine the minimal polynomial of an algebraic element over a field.