給一個有限群G，我們可不可以找到一個有理式，他的Galois group會跟G同構?
這就是Inverse Galois Problem。

在第一章我們會列出一些代數的基本定義和定理以及判別式。
第二章會找一些Sn跟An的生成源，其中在Sp有一個特殊的生成源。

第三章，我有給一個詳細的推導找出所有在S3,S4,S5的transitive subgroups。
第四章介紹一個新的群GA(p)，其中我們發現如果一個群G是transitive和solvable的話，那G就會跟GA(p)的子群共扼。

第五章我們詳細的研究三次多項式，這個在之後找三次、四次、五次不可分解的多項式很有幫助。
第六章就是找出Galois group是S3,S4,S5的transitive subgroups的例子。

Suppose given a finite group G.
Is it always possible to find a polynomial f(t) over Q such that the Galois group of f(t) over Q is isomorphic to G? This is the so-called Inverse Galois Problem.

In Section 1.1, we list some preliminary propositions and results in algebra.
In Section 1.2, we introduce the discriminant of a polynomial f(t).
This is a tool to help us to determine the Galois group of f(t) over a field K.

Note that the Galois group of an irreducible polynomial of degree n is isomorphic to a transitive subgroup in Sn.
In Chapter 2, we will study the generators of Sn and An.
There is a special case for Sp when p is a positive prime integer.
In Chapter 3, we will find all the transitive subgroups in Sn for n=3,4,5.
This is a long procedure in finding these groups.
In Chapter 4, we will introduce a special subgroups GA(p) in Sp where p is a positive prime integer. This group GA(p) is a transitive and solvable group in Sp with order p(p-1).
Moreover, a group G is a transitive and solvable subgroup of Sp if and only if G is conjugate to a subgroup of GA(p).

We let $t$ be an indeterminant over a field K in Chapter 5 and Chapter 6.
In Chapter 5, we will study the cubic polynomials f(t) over K.
We will use this result in Chapter 6.

In Chapter 6, we will find examples of irreducible polynomials whose Galois groups are transitive subgroups of Sn for n=3, 4, 5.

Introduction.......................................................................3

Chapter 1. Preliminaries...............................................5
1.1 Preliminaries............................................................5
1.2 Resultant and Discriminant.....................................6

Chapter 2. Generating Sets for Sn and An..................15
2.1. Generating Sets for Sn...........................................15
2.2. Generating Sets for An...........................................18

Chapter 3. Transitive Subgroups of Sn.................................22
3.1. Transitive Subgroups of Sn.......................................22
3.2. Transitive Subgroups of S3.......................................25
3.3. Transitive Subgroups of S4.......................................25
3.4. Transitive Subgroups of S5.......................................30

Chapter 4. The Affine Group GA(p).....................................37
4.1. The Affine Group GA(p)...........................................37
4.2. Properties of The Affine Group...................................43

Chapter 5. Radical Solutions of Cubic Polynomials.....................52
5.1. Reduction to a Monic Cubic Polynomial with no Quadratic Term.....52
5.2. Cardan Formula...................................................53
5.3. Cubic Radical Extension..........................................56
5.4. A Result from Otto Holder........................................63

Chapter 6. The Inverse Galois Problem for Small Degree................68
6.1. Relevant Results Regarding Galois Groups.........................68
6.2. The Inverse Galois Problem for Transitive Subgroups of S3........70
6.3. The Inverse Galois Problem for Transitive Subgroups of S4........72
6.4. The Inverse Galois Problem for Transitive Subgroups of S5........85

References............................................................93

[1] Eie, Minking; Chang, Shou-Te, A Course on Abstract Algebra, World Scientific, 2010. ISBN 978-981-4271-88-2.

[2] Jean-Pierre Escofier, Galois Theory, Springer, 2000. ISBN 0-387-98765-7,195-269.

[3] Nathan Jacobson, Basic Algebra, W. H. Freeman and Company, New York, 1985. ISBN 0-7167-1480-9.

[4] Bennett Kanuka, Galois Groups of Irreducible Quintic Polynomials, 2012.