The Fundamental Theorem of Algebra

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 Euler’s Identity.- 2.5 DeMoivre’s 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 Cauchy’s Theorem.- 5.1 Line Integrals and Green’s Theorem.- 5.2 Complex Integration and Cauchy’s Theorem.- 5.3 The Cauchy Integral Formula and Cauchy’s Estimate.- 5.4 Liouville’s 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 Application—The 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.- 7.8Algebraic Extensions of ? and Concluding Remarks.- Exercises.- 8 7bpology and Topological Spaces.- 8.1 Winding Number and Proof Five.- 8.2 Tbpology—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 Zbpology and the Final Proof.- 9.1 Algebraic lbpology.- 9.2 Some Further Group Theory—Abelian 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 Gauss’s Original Proof.- Appendix B: Cauchy’s 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.