Contents
1 The theory of equations 3
1.1 Primitive question . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Quadratic equations . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 The remainder theorem . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Relation between coecients and roots . . . . . . . . . . . . . 5
1.5 Complex roots of 1 . . . . . . . . . . . . . . . . . . . . . . . . 7
1.6 Cubic equations . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.7 Quartic equations . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.8 The quintic is insoluble . . . . . . . . . . . . . . . . . . . . . . 11
1.9 Prerequisites and books . . . . . . . . . . . . . . . . . . . . . 13
Exercises to Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Rings and elds 18
2.1 Denitions and elementary properties . . . . . . . . . . . . . . 18
2.2 Factorisation in Z . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Factorisation in k[x] . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Factorisation in Z[x], Eisenstein's criterion . . . . . . . . . . . 28
Exercises to Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . 32
3 Basic properties of eld extensions 35
3.1 Degree of extension . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Applications to ruler-and-compass constructions . . . . . . . . 40
3.3 Normal extensions . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4 Application to nite elds . . . . . . . . . . . . . . . . . . . . 51

3.5 Separable extensions . . . . . . . . . . . . . . . . . . . . . . . 53
Exercises to Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . 56
4 Galois theory 60
4.1 Counting eld homomorphisms . . . . . . . . . . . . . . . . . 60
4.2 Fixed subelds, Galois extensions . . . . . . . . . . . . . . . . 64
4.3 The Galois correspondences and the Main Theorem . . . . . . 68
4.4 Soluble groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.5 Solving equations by radicals . . . . . . . . . . . . . . . . . . . 76
Exercises to Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . 80
5 Additional material 84
5.1 Substantial examples with complicated Gal(L=k) . . . . . . . 84
5.2 The primitive element theorem . . . . . . . . . . . . . . . . . 84
5.3 The regular element theorem . . . . . . . . . . . . . . . . . . . 84
5.4 Artin{Schreier extensions . . . . . . . . . . . . . . . . . . . . . 84
5.5 Algebraic closure . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.6 Transcendence degree . . . . . . . . . . . . . . . . . . . . . . . 85
5.7 Rings of invariants and quotients in algebraic geometry . . . . 86
5.8 Thorough treatment of inseparability . . . . . . . . . . . . . . 86
5.9 AOB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.10 The irreducibility of the cyclotomic equation . . . . . . . . . . 86
Exercises to Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . 87