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140122 ||| eng |
| 020 |
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|a 9781461219286
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| 100 |
1 |
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|a Fine, Benjamin
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| 245 |
0 |
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|a The Fundamental Theorem of Algebra
|h Elektronische Ressource
|c by Benjamin Fine, Gerhard Rosenberger
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| 250 |
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|a 1st ed. 1997
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| 260 |
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|a New York, NY
|b Springer New York
|c 1997, 1997
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| 300 |
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|a XI, 210 p
|b online resource
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| 505 |
0 |
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|a 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 --
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| 505 |
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|a 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 --
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| 505 |
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|a 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
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| 653 |
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|a Mathematical analysis
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| 653 |
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|a Analysis
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| 653 |
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|a Topology
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| 653 |
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|a Algebra
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| 700 |
1 |
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|a Rosenberger, Gerhard
|e [author]
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| 041 |
0 |
7 |
|a eng
|2 ISO 639-2
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| 989 |
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|b SBA
|a Springer Book Archives -2004
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| 490 |
0 |
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|a Undergraduate Texts in Mathematics
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| 028 |
5 |
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|a 10.1007/978-1-4612-1928-6
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| 856 |
4 |
0 |
|u https://doi.org/10.1007/978-1-4612-1928-6?nosfx=y
|x Verlag
|3 Volltext
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| 082 |
0 |
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|a 512
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