The purpose of this book is to examine three pairs of proofs of the theorem from three different areas of mathematics: abstract algebra, complex analysis and topology. It is ideal for a "capstone" course in mathematics for junior/senior level undergraduate mathematics students or first year graduate students. It could also be used as an alternative approach to an undergraduate abstract algebra course.
Inhaltsverzeichnis
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.
