Fundamental Theorem of Algebra

AvBenjamin Fine,Gerhard Rosenberger

Inbunden, Engelska, 1997

Del i serien Undergraduate Texts in Mathematics

870 kr

Beställningsvara. Skickas inom 10-15 vardagar. Fri frakt över 249 kr.

Fler format och utgåvor

Häftad

622 kr

Beskrivning

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.

Produktinformation

Utgivningsdatum:1997-06-20
Mått:155 x 235 x 27 mm
Vikt:503 g
Format:Inbunden
Språk:Engelska
Serie:Undergraduate Texts in Mathematics
Antal sidor:210
Upplaga:1997
Förlag:Springer-Verlag New York Inc.
ISBN:9780387946573

Utforska kategorier

Innehållsförteckning

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.

Företag, bibliotek eller offentlig verksamhet?

Fundamental Theorem of Algebra

Fler format och utgåvor

Beskrivning

Produktinformation

Utforska kategorier

Innehållsförteckning

Mina sidor

Mina sidor

Hjälp

Hjälp

Om Bokus

Om Bokus

Populärt

Populärt

Inspiration

Inspiration

Fundamental Theorem of Algebra

Fler format och utgåvor

Beskrivning

Produktinformation

Utforska kategorier

Innehållsförteckning