## Table of Contents

**Preface**

**1. Introduction to Galois’ Proof **

1.1 Radicals and symmetry

1.2 Book outline

**2. Theorems related to symmetry **

2.1 Vandermonde Determinants

2.2 Symmetric expressions

2.3 Resolvents

**3. Euclid’s algorithm and shared roots**

3.1 Euclid’s algorithm

3.2 Roots shared with an irreducible polynomial equation

**4. Galois’ Proof, Stage 1**

4.1 The Resolvent Equation

4.2 Introducing the Galois Group (First Stage of Galois’ Proof, Part 1)

4.3 The Galois set of arrangements (First Stage of Galois’ Proof, Part 2)

4.4 Permutations, presentations and groups

4.4.1 Permutations and permutation notations

4.4.2 Permutation composition

4.4.3 Presentations of a group

4.4.4 Column permutations

4.5 Groups and Subgroups, Modern and Galois

4.6 Presentations and subgroups

4.7 Normal subgroups

4.8 Proof that the Galois’ set defines a group (First Stage of Galois’ Proof, Part 3)

4.8.1 Galois’ Proposition 1

4.9 Automorphisms

4.10 Continuing the proof that the Galois’ set defines a group (First Stage of Galois’ Proof, Part 4)

**5. Galois’ Proof, Stage 2**

5.1 Fields of numbers

5.2 Roots of unity

5.3 Expanding the field of numbers using radicals (Second Stage of Galois’ Proof, Part 1)

5.4 Using alternative roots of unity

5.5 Adjoining radicals implies normal subgroups Second Stage of Galois’ Proof, Part 2

5.6 Indirect Permutation Specification

5.7 Completing Galois’ Proposition 2 (Second Stage of Galois’ Proof, Part 3)

**6. Final steps to the impossible quintic**

6.1 Listing the steps

6.2 Proof the Galois group of irreducible quintics contains a 5-cycle

6.2.1 Transitivity

6.2.2 Transitivity implies a 5 cycle

6.3 Proof that some Galois groups associated with irreducible quintics contain a 2-cycle: Permutations of Complex Conjugacy

6.4 Group generated by a 5 cycle and a 2 cycle

6.5 The group of all permutations of 5 objects S5, does not have a normal subgroup of index 5

6.6 Specifying an irreducible quintic with exactly two complex roots

6.7 Factorisation over rationals

6.8 An insoluble quintic

**7. Summary**

**Appendix**