1 Introduction and Historical Remarks -- 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 Eulerโs Identity -- 2.5 DeMoivreโs Theorem for Powers and Roots -- Exercises -- 3 Polynomials and Complex Polynomials -- 3.1 The Ring 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 Cauchyโs Theorem -- 5.1 Line Integrals and Greenโs Theorem -- 5.2 Complex Integration and Cauchyโs Theorem -- 5.3 The Cauchy Integral Formula and Cauchyโs Estimate -- 5.4 Liouvilleโs Theorem and the Fundamental Theorem of Algebra: Proof Ttvo -- 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 ApplicationโThe Transcendence of e and ? -- 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 ? and Concluding Remarks -- Exercises -- 8 7bpology and Topological Spaces -- 8.1 Winding Number and Proof Five -- 8.2 TbpologyโAn Overview -- 8.3 Continuity and Metric Spaces -- 8.4 Topological Spaces and Homeomorphisms -- 8.5 Some Further Properties of Topological Spaces -- Exercises -- 9 Algebraic Zbpology and the Final Proof -- 9.1 Algebraic lbpology -- 9.2 Some Further Group TheoryโAbelian 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 Gaussโs Original Proof -- Appendix B: Cauchyโs Theorem Revisited -- Appendix C: Three Additional Complex Analytic Proofs of the Fundamental Theorem of Algebra -- Appendix D: Two More Ibpological Proofs of the Fundamental Theorem of Algebra -- Bibliography and References

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