Grobner basis examination on fields and some modules

In this thesis, Gröbner bases were examined and some information was given about how to make polynomial reduction by using Gröbner basis in modules. In the first part of the thesis, introduction was made and reduction of univariate and multivariate polynomials, term ordering and division algorithm over fields are emphasized in the second section. The third section is attributed to the Gröbner basis definition, S-Polynomials, Buchberger Algorithm and Reduced Gröbner bases. Finally, in the fourth chapter, some information about Module theory is given and Gröbner bases in Modules are discussed.