The Calculation Of The Isomorphism Classes Of Abelian Groups And Groebner Basis Over Valuation Rings

In the first part of this paper, we give a computational formula to calculate the isomorphism classes of a finite abelian group, by using the partition theory of integers in combinatorics and the fundamental structure theorem of finite abelian groups, so that one could determine easily isomorphism classes of any finite abelian group once its order is given. Furthermore, we also implement the algorithm on computer, which will automate the computation.The second part of this paper focuses on studing some properties of Groebner basis on the general valuation rings and introducing the application of Groebner bases in the code theory. The concept of Groebner basis was first proposed by Buchberger to solve the ideal membership problem for polynomial rings over domains, since then the method has been extensively researched and developed, and widely used in automated proof of geometric theorems, solving algebraic equations and the problems in graph theory, computational algebraic number theory, geometry and commutative algebra, cryptography and coding theory, integer programming, image processing, etc. In this part, we firstly study the relationship between the normal form and Groebner basis for a polynomial ring over a general valuation ring, and find out the algorithm of the normal form, such that we can through the normal form determine that if a subset is a Groebner basis of an ideal; then we generalize the concepts of Gauss basis and Gauss representation from the field to the valuation ring, and present the relationship between S-polynomials and Groebner basis over a general valuation ring; lastly, we introduce an application of Groebner basis on coding theory, computing the dimension of the code through Groebner basis.The thesis is divided into four chapters:The first chapter briefly introduces the structure of a finitely generated Abelian group and the overall development of Groebner basis theory.The second chapter gives a computational formula to calculate the isomorphism classes of a finite abelian group, by using the partition theory of integers in combinatorics and the fundamental structure theorem of finite abelian groups.The third chapter studing the relationship between normal form, S-polynomial and Groebner basis over general valuation rings.The last chapter briefly introduces some application of Groebner basis in coding theory:computing the code dimension through Groebner basis.