Fields and Galois Theory

✍ Scribed by John M. Howie

• Publisher: Springer
• Year: 2006
• Tongue: English
• Leaves: 234
• Series: Springer Undergraduate Mathematics Series
• Edition: 1
• Category: Library

✦ Synopsis

The pioneering work of Abel and Galois in the early nineteenth century demonstrated that the long-standing quest for a solution of quintic equations by radicals was fruitless: no formula can be found. The techniques they used were, in the end, more important than the resolution of a somewhat esoteric problem, for they were the genesis of modern abstract algebra.

This book provides a gentle introduction to Galois theory suitable for third- and fourth-year undergraduates and beginning graduates. The approach is unashamedly unhistorical: it uses the language and techniques of abstract algebra to express complex arguments in contemporary terms. Thus the insolubility of the quintic by radicals is linked to the fact that the alternating group of degree 5 is simple - which is assuredly not the way Galois would have expressed the connection.

✦ Table of Contents

Preface......Page 7
Contents......Page 9
1.1 Definitions and Basic Properties......Page 11
1.2 Subrings, Ideals and Homomorphisms......Page 15
1.3 The Field of Fractions of an Integral Domain......Page 23
1.4 The Characteristic of a Field......Page 27
1.5 A Reminder of Some Group Theory......Page 30
2.1 Euclidean Domains......Page 35
2.2 Unique Factorisation......Page 39
2.3 Polynomials......Page 43
2.4 Irreducible Polynomials......Page 51
3.1 The Degree of an Extension......Page 61
3.2 Extensions and Polynomials......Page 64
3.3 Polynomials and Extensions......Page 74
4.1 Ruler and Compasses Constructions......Page 81
4.2 An Algebraic Approach......Page 84
5. Splitting Fields......Page 89
6. Finite Fields......Page 95
7.1 Monomorphisms between Fields......Page 101
7.2 Automorphisms, Groups and Subfields......Page 104
7.3 Normal Extensions......Page 113
7.4 Separable Extensions......Page 119
7.5 The Galois Correspondence......Page 125
7.6 The Fundamental Theorem......Page 129
7.7 An Example......Page 134
8.1 Quadratics, Cubics and Quartics: Solution by Radicals......Page 137
8.2 Cyclotomic Polynomials......Page 143
8.3 Cyclic Extensions......Page 150
9.1 Abelian Groups......Page 159
9.2 Sylow Subgroups......Page 165
9.3 Permutation Groups......Page 170
9.4 Properties of Soluble Groups......Page 177
10. Groups and Equations......Page 179
10.1 Insoluble Quintics......Page 183
10.2 General Polynomials......Page 184
10.3 Where Next......Page 190
11.1 Preliminaries......Page 193
11.2 The Construction of Regular Polygons......Page 197
12. Solutions......Page 203
Bibliography......Page 229
List of Symbols......Page 231
Index......Page 233