在這篇論文中，我們將會看到一些field extension 與Galois群的主要定理與定義。在第三章中，我們將會考慮三次、四次方程式的性質特性，並且考慮一些三次、四次的特殊方程式與所有一般方程式的Galois群的解法。在第五章中，我們會先討論一般五次方程式的判別式，然後再試著去分析五次方程式的Galois群與判別式有什麼結構與特性。

In the thesis, we will see that some main theories about the field extension and Galois groups.In chapter 3, we consider some propositions about the polynomial of degree 3, 4 and apply them to solve the Galois groups of some special polynomials or general polynomials.In chapter 5, we consider the discriminant of the general qunitic polynomial and try to analysis the structure of the Galois group about the discriminant.

1.Introduction……………………………………p.1 2.Background………………………………………p.2 
3.The polynomial of degree 3 and 4…………p.15
4.Some results about the Galois group……p.26
5.On Galois group………………………………p.39
6.References……………………………………p.45

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