| preface 1 introduction and historical remarks 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 king 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 5 complex integration and cauchy's theorem 6 fields and field extensions 7 galois theory 8 topology and topological spaces algebraic topology and the final proof 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 topological proofs of the fundamental theorem of algebra bibliography and references index |
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