Galois Theory

Cover -- Half Title -- Title Page -- Copyright Page -- Dedication -- Contents -- Acknowledgements -- Preface to the Fifth Edition -- Historical Introduction -- 1. Classical Algebra -- 1.1. Complex Numbers -- 1.2. Subfields and Subrings of the Complex Numbers -- 1.3. Solving Equations -- 1.4. Solution by Radicals -- 2. The Fundamental Theorem of Algebra -- 2.1. Polynomials -- 2.2. Fundamental Theorem of Algebra -- 2.3. Implications -- 3. Factorisation of Polynomials -- 3.1. The Euclidean Algorithm -- 3.2. Irreducibility -- 3.3. Gauss's Lemma -- 3.4. Eisenstein's Criterion -- 3.5. Reduction Modulo p -- 3.6. Zeros of Polynomials -- 4. Field Extensions -- 4.1. Field Extensions -- 4.2. Rational Expressions -- 4.3. Simple Extensions -- 5. Simple Extensions -- 5.1. Algebraic and Transcendental Extensions -- 5.2. The Minimal Polynomial -- 5.3. Simple Algebraic Extensions -- 5.4. Classifying Simple Extensions -- 6. The Degree of an Extension -- 6.1. Definition of the Degree -- 6.2. The Tower Law -- 6.3. Primitive Element Theorem -- 7. Ruler-and-Compass Constructions -- 7.1. Approximate Constructions and More General Instruments -- 7.2. Constructions in C -- 7.3. Specific Constructions -- 7.4. Impossibility Proofs -- 7.5. Construction from a Given Set of Points -- 8. The Idea behind Galois Theory -- 8.1. A First Look at Galois Theory -- 8.2. Galois Groups According to Galois -- 8.3. How to Use the Galois Group -- 8.4. The Abstract Setting -- 8.5. Polynomials and Extensions -- 8.6. The Galois Correspondence -- 8.7. Diet Galois -- 8.8. Natural Irrationalities -- 9. Normality and Separability -- 9.1. Splitting Fields -- 9.2. Normality -- 9.3. Separability -- 10. Counting Principles -- 10.1. Linear Independence of Monomorphisms -- 11. Field Automorphisms -- 11.1. K-Monomorphisms -- 11.2. Normal Closures -- 12. The Galois Correspondence.