在這篇論文中，我們將會學習Groebner基底和不變量環的一些基本性質。我們將給予一些例題去說明如何利用Groebner基底來解聯立方程式，並將Groebner基底應用在理想元素的隸屬問題以及求兩個理想的交集。我們也將它使用在不變量環的一些性質，這些性質主要描述說當給定一個有限群後，它的不變量環生成元可利用 Groebner基底來求出關係。
    這篇論文主要分成三節。第一節是探討有關Groebner基底的基本性質，並且討論當在解聯立方程式時所使用到的消去和擴展定理。第二節探討當給定任何一個有限群時我們如何創造出它的不變量環。在最後一節，我們將給一些可以使用不變量環的例子。第一個例子，我們探討一個立方體於對稱群的所有不變量。而第二個例子我們將會使用電腦來幫我們求出它的不變量環。這篇論文裡我們使用Maple來幫我們計算。

In this thesis, we shall study some  basic properties of Groebner basis and ring of invariants. We shall give some examples to show the use of Groebner basis in solving polynomial equation, determining ideal memberships and intersection of ideals. We shall also discuss basic theories on ring of invariant. Some examples on the construction of generators for a ring of invariant of finite group will be given. Relations among the generators will be found by using  Groebner basis.
   This thesis is divided into three sections. Section 1 is about basic properties of Groebner basis, then talk about elimination and extension theorems which is used in solving
polynomial equation. Section 2 is about how to construct the ring of invariant when giving a finite group. In the last section, we shall give some examples on calculation of ring of invariant.  In particular, we discuss the invariants of the symmetry group of a cube. We also
give an example to show how to compute explicitly ring of invariant of an abelian group.  Groebner basis calculation is implemented in many computer algebra systems, such as Maple, MATHEMATICA, Cocoa etc.  In this thesis, we use Maple to do our calculations.

目錄
1 Groebner基底-----------------------------------------------------------------------------------------------2
  1.1 Groebner基底的性質-------------------------------------------------------------------------------------2
  1.2 消去以及擴展定理---------------------------------------------------------------------------------------6

2 不變量環---------------------------------------------------------------------------------------------------9
  2.1 不變量環的性質-----------------------------------------------------------------------------------------9
  2.2 構出不變量環的生成物-------------------------------------------------------------------------------13

3 一些不變量環的例子-------------------------------------------------------------------------------------18

Contents
1 Groebner Basis-------------------------------------------------------------------------------------------26
  1.1 The properties of Groebner basis-----------------------------------------------------------------26
  1.2 The elimination and extension theorem---------------------------------------------------------33

2 Ring of Invariant-----------------------------------------------------------------------------------------42
  2.1 The properties of the ring of invariant-------------------------------------------------------------42
  2.2 Constructing generators for the ring of invariant----------------------------------------------47

3 Some Example of the Ring of Invariant-------------------------------------------------------------53

References----------------------------------------------------------------------------------------------------61

[1] D.Cox, F.Little, D.O'shea, Ideal, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Springer-Verlag, 1992

[2] Mara D.Neusel, Invariant Theory,American Mathematical Society(AMS), 2006

[3] B.sturmfels, Algorithm in invariant Theory: Text and Monographs in symbolic computation, Springer-Verlag, 1993

[4] Thomas W, Hungerford, Algebra, Springer-Verlag, 1974

[5] W.Keith Nicholson, Introduction to abstract algebra, Wiley, 1999

[6] David S.Dummit, Richard M.Foote, Abstract algebra, Prentice-Hall, 1991

[7] Hideyuki Matsumura, Commutative ring theory,Cambridge University Press, 1986