The Fundamental Theorem of AlgebraThe fundamental theorem of algebra states that any complex polynomial must have a complex root. This book examines three pairs of proofs of the theorem from three different areas of mathematics: abstract algebra, complex analysis and topology. The first proof in each pair is fairly straightforward and depends only on what could be considered elementary mathematics. However, each of these first proofs leads to more general results from which the fundamental theorem can be deduced as a direct consequence. These general results constitute the second proof in each pair. To arrive at each of the proofs, enough of the general theory of each relevant area is developed to understand the proof. In addition to the proofs and techniques themselves, many applications such as the insolvability of the quintic and the transcendence of e and pi are presented. Finally, a series of appendices give six additional proofs including a version of Gauss'original first proof. The book is intended for junior/senior level undergraduate mathematics students or first year graduate students, and would make an ideal "capstone" course in mathematics. |
Contents
II | 1 |
IV | 5 |
V | 10 |
VI | 12 |
VII | 14 |
VIII | 17 |
IX | 19 |
X | 21 |
XLII | 102 |
XLIII | 104 |
XLVI | 105 |
XLVII | 112 |
XLVIII | 115 |
XLIX | 119 |
L | 123 |
LI | 124 |
XI | 24 |
XII | 27 |
XIII | 29 |
XIV | 31 |
XV | 33 |
XVI | 34 |
XVII | 36 |
XX | 41 |
XXI | 46 |
XXII | 49 |
XXIII | 52 |
XXVI | 61 |
XXVII | 66 |
XXVIII | 70 |
XXIX | 71 |
XXX | 72 |
XXXII | 74 |
XXXV | 81 |
XXXVI | 84 |
XXXVII | 86 |
XXXVIII | 91 |
XXXIX | 94 |
XLI | 99 |
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Common terms and phrases
a₁ abelian group algebraic over F algebraically closed analytic automorphism calculus called Cauchy-Riemann equations Cauchy's theorem Chapter circle coefficients complex function complex number complex polynomial complex root conjugate constructible continuous function Corollary coset cyclic group define Definition deg P(x denoted differentiable domain elementary symmetric polynomials elements EXAMPLE exists extension of F f(zo factors field extension field F fixed field follows free abelian Fundamental Theorem g₁ Gal(K/F Galois extension Galois group Galois theory Green's theorem H₁ hence homeomorphic homology homotopic irr(a irreducible polynomial isomorphism Lemma Let f(z line integral linear loop metric space multiplication nonconstant nonzero nth root open sets permutation PoP1 proof properties Prove real polynomial region splitting field subfield subgroup subset Suppose f(z t₁ Theorem of Algebra topological space triangle vector space winding number y₁ zero ди მა
References to this book
Proofs and Fundamentals: A First Course in Abstract Mathematics Ethan D. Bloch No preview available - 2000 |
The Classical Fields: Structural Features of the Real and Rational Numbers H. Salzmann Limited preview - 2007 |