Gröbner bases and its applications on computer

This thesis includes the analysis of the algebraic structure of Gröbner's basis theory as well as its use in different programming languages. In the first chapter, there is the introduction part in which the historical development process of Gröbner basis theory is explained. In the second part, the basic definitions and features required for the thesis work are given. In the third chapter, firstly the Gröbner basis concept is introduced and its characterizations are given, then the Buchberger algorithm, which is a method for calculating the Gröbner basis, is explained with examples. In addition, it has been shown that two different Gröbner bases can be written under the name of a minimum and reduced from a Gröbner base. In the fourth chapter, how the Gröbner base is calculated in some programming languages is explained and examples are given.