Preface
1 Introduction and Historical Remarks Complex Numbers
2 Complex Numbers
2.1 Fields and the Real Field
2.2 The Complex Number Field
2.3 Geometrical Representation of Complex Numbers
2.4 Polar Form and Eulers Identity
2.5 DeMoivres Theorem for Powers and Roots Exercises
3 Polynomials and Complex Polynomials
3.1 The King of Polynomials over a Field
3.2 Divisibility and Unique Factorization of Polynomials
3.3 Roots of Polynomials and Factorization
3.4 Real and Complex Polynomials
3.5 The Fundamental Theorem of Algebra： Proof One
3.6 Some Consequences of the Fundamental Theorem Exercises
4 Complex Analysis and Analytic Functions
4.1 Complex Functions and Analyticity
4.2 The Cauchy-Riemann Equations
4.3 Conformal Mappings and Analyticity
Exercises
5 Complex Integration and Cauchys Theorem
5.1 Line Integrals and Greens Theorem
5.2 Complex Integration and Cauchys Theorem
5.3 The Cauchy Integral Formula and Cauchys Estimate
5.4 Liouviues Theorem and the Fundamental Theorem of Algebra： Proof Two
5.5 Some Additional Results
5.6 Concluding Remarks on Complex Analysis
Exercises
6 Fields and Field Extensions
6.1 Algebraic Field Extensions
6.2 Adjoining Roots to Fields
6.3 Splitting Fields
6.4 Permutations and Symmetric Polynomials
6.5 The Fundamental Theorem of Algebra： Proof Three
6.6 An Application——The Transcendence of e and ~r
6.7 The Fundamental Theorem of Symmetric Polynomials
Exercises
7 Galois Theory
7.1 Galois Theory Overview
7.2 Some Results From Finite Group Theory
7.3 Galois Extensions
7.4 Automorphisms and the Galois Group
7.5 The Fundamental Theorem of Galois Theory
7.6 The Fundamental Theorem of Algebra： Proof Four
7.7 Some Additional Applications of Galois Theory
7.8 Algebraic Extensions of R and Concluding Remarks
Exercises
8 Topology and Topological Spaces
8.1 Winding Number and Proof Five
8.2 Topology——An Overview
8.3 Continuity and Metric Spaces
8.4 Topological Spaces and Homeomorphisms
8.5 Some Further Properties of Topological Spaces
Exercises
9 Algebraic Topology and the Final Proof
9.1 Algebraic Topology
9.2 Some Further Group Theory——Abclian Groups
9.3 Homotopy and the Fundamental Group
9.4 Homology Theory and Triangulations
9.5 Some Homology Computations
9.6 Homology of Spheres and Brouwer Degree
9.7 The Fundamental Theorem of Algebra： Proof Six
9.8 Concluding Remarks
Exercises
Appendix A： A Version of Gausss Original Proof
Appendix B： Cauchys Theorem Revisited
Appendix C： Three Additional Complex Analytic Proofs of the Fundamental Theorem of Algebra
Appendix D： Two More Topological Proofs of the Fundamental Theorem of Algebra
Bibliography and References
Index
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