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1. Rings and Fields ............................................ 1
1.1  Definitions and  Basic Properties ............................  1
1.2  Subrings, Ideals and Homomorphisms .......................  5
1.3 The Field of Fractions of an Integral Domain ................ 13
1.4 The Characteristic of a Field ............................ 17
1.5 A Reminder of Some Group Theory......................... 20
2. Integral Domains and Polynomials .......................... 25
2.1  Euclidean Domains .................  ................... .  25
2.2 Unique Factorisation .................................... 29
2.3  Polynomials  ..................... ........... ........ .  33
2.4  Irreducible Polynomials  .............................. . .   .  41
3. Field Extensions ......................................... 51
3.1 The Degree of an Extension. ............................  51
3.2  Extensions and  Polynomials  ...............................  54
3.3  Polynomials and  Extensions  ...............................  64
4. Applications to Geometry .............................. . . 71
4.1  Ruler and Compasses Constructions  ........................  71
4.2  An  Algebraic Approach  .............................. . .   .  74
5. Splitting Fields ................ ......................... 79
6. Finite Fields ................ ................................ 85
7. The Galois Group .......................................... 91
7.1 Monomorphisms between Fields ............................ 91
7.2 Automorphisms, Groups and Subfields ...................... 94
7.3 Normal Extensions ............... ................... 103
7.4  Separable Extensions  ............... ................. .   . 109
7.5 The Galois Correspondence . ............................. 115
7.6 The Fundamental Theorem ............................ .  . 119
7.7  An  Example  ............................................  124
8. Equations and Groups ................ .................... 127
8.2 Cyclotomic Polynomials . ............................... 133
8.3 Cyclic Extensions ........................................140
9.  Some  Group  Theory  .........................  .............149
9.1  Abelian  Groups  ......................................... 149
9.2  Sylow  Subgroups  ........................................ 155
9.3 Permutation Groups ................  ................... 160
9.4 Properties of Soluble Groups ............. .............. 167
10.  Groups  and  Equations  ...................................... 169
10.1 Insoluble Quintics ............... ................... . 173
10.2 General Polynomials ................  ................... 174
10.3 Where Next? ......................................... 180
11. Regular Polygons ..........................................  183
11.1 Preliminaries ............  ............................183
11.2 The Construction of Regular Polygons ...................... 187
12. Solutions ....................................    ...........193

Library of Congress subject headings for this publication: Galois theory, Algebraic fields