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The FUNDAMENTAL THEOREM OF ALGEBRA states that any complex polynomial must have a complex root. This basic result, whose first accepted proof was given by Gauss, lies really at the intersection of the theory of numbers and the theory of equations, and arises also in many other areas of mathematics. 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. 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 lends itself to generalizations, which in turn, lead to more general results from which the fundamental theorem can be deduced as a direct consequence. These general results constitute the second prooof 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 trascendence 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. It is ideal for a "capstone" course in mathematics. It could also be used as an alternative approach to an undergraduate abstract algebra course. Finally, because of the breadth of topics it covers it would also be ideal for a graduate course for mathmatics teachers. ... Read more

Customer Reviews (3)

Will please some and disappoint others
This book will please some and disappoint others, depending on what you seek. Incidentally it supplies 12 proofs of the theorem including Gauss' original proof in his doctoral thesis, half of these proofs being in the appendices.

The authors do a superb job of showing how disparate areas of mathematics can be brought to bear on a single problem from different directions. Many problems would have been suitable for such a purpose but the Fundamental Theorem of Algebra is particularly well chosen due to its importance and the fact that none of the proofs are long.

Yet despite my eagerness to read this book I was disappointed. While utilizing disparate areas of mathematics I wanted more insight into the theorem from almost every one of the proofs. The best proof I have seen of this theorem is in an appendix of G. H. Hardy's book A Course of Pure Mathematics. That proof gives insight as to why the theorem is true because it is a constructive proof (it actually constructs a root) that is easily understood, something not true of all constructive proofs. I was surprised to find Hardy's proof not present in this book in this form but only in a much weaker form (proof five) that is not constructive.

Asking for such clarity of insight is asking too much of most proofs of this theorem but I wanted more in that direction. As a specific example, this theorem can not be proved without appealing somewhere to continuity, which makes the proof itself partake of analysis and not purely algebra. The book does not point out in every proof (e.g., proof four) where such an appeal is made and I suspect it will often be hidden from students who may not grasp that continuity is an essential.

Nevertheless I did enjoy this book and definitely recommend it for the stated purpose of showing how disparate areas of mathematics can be applied to the same problem, for it accomplishes this purpose well. I applaud the authors for a superb effort although, like a literary critic reviewing a play but who can not write one, I wish there had been more. And of course, the authors may read my comments with astonishment and disbelief.

Finally, both mathematical comments in the two-star review by "A Customer" are incorrect; the book is actually correct on these points nor did I find others errors in it.

Too Poorly Written
To be sure, the aim of this book is interesting, but it's too poorly written. There are so many serious mistakes that I wonder why the students who took the course of these authors didn't point them out.
For example, the latter half of Lemma 4.1.5 is wrong for it lacks (-i), and Theorem 7.3.3 is wrong for |K:F| is not equal to the number of automorphisms of K fixing F, but is equal to the number of injective isomorphisms form K to algebraic closure of F fixing F.
I recommend you to read more reliable books.

A wonderful text!
This text gives a number of proofs of the Fundamental Theorem of Algebra (FTA).The proofs come from diverse areas of mathematics including complex analysis, abstract algebra (Galois theory) and topology.A version ofGauss's original proof is even given.The background information leadingup to each proof makes the book readable for a math major who has taken afirst course in abstract algebra.A student who uses this text will haveseen a lot of really interesting mathematics by the end and will have asense of how the mathematics can be used. ... Read more

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Chapters: Vector Space, Group, Linear Map, Polynomial, Euclidean Vector, Cauchy Sequence, Fundamental Theorem of Algebra, Power Set, Dimension, Binary Operation, Isomorphism, Tensor Product, Monoid, Universal Algebra, Algebraic Topology, Algebraic Number, Linear Combination, Linear Independence, Identity Element, Union, Permutation, Algebraic Structure, Formal Power Series, Idempotence, Associativity, Algebraically Closed Field, List of Group Theory Topics, Linear Span, Groupoid, Inverse Limit, Binary Function, Fundamental Theorem on Homomorphisms, Direct Product, Arity, Commutator, Unique Factorization Domain, Isomorphism Class, General Linear Group, Zero Divisor, Embedding, Distributivity, Magma, Homological Algebra, Row and Column Spaces, Complement, Inverse Element, Identity Matrix, Cross Product, Orthogonality, Special Linear Group, Additive Inverse, Eigenvalue, Eigenvector and Eigenspace, List of Algebraic Structures, Algebraic Number Field, Special Classes of Semigroups, Emmy Noether Bibliography, Problems in Loop Theory and Quasigroup Theory, Inverse Semigroup, Interior Algebra, Order of Operations, Light's Associativity Test, Information Algebra, Algebraic K-Theory, Module, Operad Theory, Seifert-van Kampen Theorem, Esquisse D'un Programme, Hecke Algebra, Commutativity, Cycle Graph, Zero-Product Property, Free Object, Valuation Ring, Higher-Dimensional Algebra, Multiplicative Inverse, Pseudo-Ring, Group Extension, Semiring, Semigroup With Two Elements, Cancellative Semigroup, Transpose, Direct Limit, Grothendieck Group, Closure With a Twist, Irreducible Polynomial, Polarization Identity, Involution, List of Abstract Algebra Topics, Variety, Scalar, Absolute Value, Double Groupoid, Cokernel, Dimension Theorem for Vector Spaces, Intersection, Homomorphic Secret Sharing, Near-Field, Racks and Quandles, Goodman-nguyen-van Fraassen Algebra, Conditional Event Algebra, Nearring, Pseudogroup, Anticommutativity, Composition R...More: http://booksllc.net/?id=32370 ... Read more

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Chapters: Fundamental Theorem of Algebra, Fundamental Theorem of Arithmetic, Finitely Generated Abelian Group, Fundamental Theorem on Homomorphisms, Thermodynamic Potential, Fundamental Theorem of Calculus, Fundamental Theorems of Welfare Economics, Fundamental Theorem of Combinatorial Enumeration, Fundamental Theorem of Galois Theory, Helmholtz Decomposition, Collineation, Fundamental Theorem of Cyclic Groups, Fundamental Theorem of Riemannian Geometry, Fisher's Fundamental Theorem of Natural Selection, Fundamental Theorem of Poker, Fundamental Lemma of Calculus of Variations, Fundamental Theorem of Linear Algebra, Fundamental Theorem of Arbitrage-Free Pricing, Fundamental Theorem of Curves, Fundamental Theorem of Linear Programming. Source: Wikipedia. Pages: 109. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: In mathematics, the fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. Equivalently, the field of complex numbers is algebraically closed. Sometimes, this theorem is stated as: every non-zero single-variable polynomial with complex coefficients has exactly as many complex roots as its degree, if each root is counted up to its multiplicity. Although this at first appears to be a stronger statement, it is a direct consequence of the other form of the theorem, through the use of successive polynomial division by linear factors. In spite of its name, there is no purely algebraic proof of the theorem, since any proof must use the completeness of the reals (or some other equivalent formulation of completeness), which is not an algebraic concept. Additionally, it is not fundamental for modern algebra; its name was given at a time in which algebra was mainly about solving polynomial equations with...More: http://booksllc.net/?id=51414 ... Read more

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Chapters: Field, P-Adic Number, Fundamental Theorem of Algebra, Hyperreal Number, Galois Theory, Finite Field, Algebraically Closed Field, Galois Group, Field of Fractions, Local Field, Number System, Algebraic Number Field, Valuation, Rational Number, Archimedean Property, Real Closed Field, Fundamental Theorem of Galois Theory, Glossary of Field Theory, Eisenstein's Criterion, Tensor Product of Fields, Dirichlet's Unit Theorem, Cubic Field, Rational Variety, P-Adically Closed Field, Exponential Field, Iwasawa Theory, Characteristic, Transcendence Degree, Construction of Splitting Fields, Quadratic Field, Kummer Theory, Differential Galois Theory, Polynomial Basis, Perfect Field, Primitive Polynomial, Thin Set, Primitive Element Theorem, Stark Conjectures, Additive Polynomial, Brumer-stark Conjecture, Grothendieck's Galois Theory, Separable Polynomial, Quasi-Finite Field, Generic Polynomial, Quasi-Algebraically Closed Field, Global Field, Serre's Conjecture Ii, Field Norm, Multiplicative Group, Embedding Problem, Conjugate Element, Pseudo Algebraically Closed Field, Levi-Civita Field, Tschirnhaus Transformation, List of Number Fields With Class Number One, Normal Basis, Minimal Polynomial, Adjunction, Superreal Number, Field Trace, Change of Bases, Totally Real Number Field, Cm-Field, Krasner's Lemma, All One Polynomial, Formally Real Field, Strassmann's Theorem, Norm Form, Equivariant L-Function, Function Field Sieve, Isomorphism Extension Theorem, Euclidean Field, Composite Field, Equally Spaced Polynomial, Complete Field. Source: Wikipedia. Pages: 335. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: In mathematics, an algebraic number field (or simply number field) F is a finite (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q ...More: http://booksllc.net/?id=223788 ... Read more

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In mathematics, the fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. Equivalently, the field of complex numbers is algebraically closed. Sometimes, this theorem is stated as: every non-zero single-variable polynomial, with complex coefficients, has exactly as many complex roots as its degree, if each root is counted up to its multiplicity. Although this at first appears to be a stronger statement, it is a direct consequence of the other form of the theorem, through the use of successive polynomial division by linear factors. ... Read more

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In mathematics, physics, and engineering, the imaginary unit allows the real number system, mathbb{R}, to be extended to the complex number system, mathbb{C}.It is denoted by i or the Latin j or the Greek iota (see Alternative notations below). Its precise definition is dependent upon the particular method of extension. The primary motivation for this extension is the fact that not every polynomial equation with real coefficients f(x) = 0 has a solution in the real number system. In particular, the equation x2 + 1 = 0 has no real solution (see Definition below). However, if complex numbers are allowed as solutions, then this equation, and indeed every non-zero degree polynomial equation f(x) = 0, has a solution (see Algebraic closure and Fundamental theorem of algebra). The imaginary unit is often loosely referred to as the "square root of ?1". However, there are in fact two square roots of ?1 (just as there are two square roots of every real number), namely i and ?i. Misuse of the imaginary unit may lead to difficulties. ... Read more

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Chapters: Euler's Formula, Complex Number, Euler's Identity, Exponential Function, Polynomial, Fundamental Theorem of Algebra, Imaginary Number, Holomorphic Function, de Moivre's Formula, Cauchy's Integral Formula, Cauchy's Integral Theorem, List of Fourier-Related Transforms, Laurent Series, Residue Theorem, Riemann Mapping Theorem, Entire Function, Meromorphic Function, Cauchy-riemann Equations, Analytic Continuation, Casorati-weierstrass Theorem, Pole, Removable Singularity, Analytic Function, Winding Number, Riemann Surface, Essential Singularity, Generalized Continued Fraction, Methods of Contour Integration, Holomorphic Functional Calculus, Complex Plane, Complex Logarithm, Dessin D'enfant, Branch Point, Mellin Transform, Gauss's Continued Fraction, Almost Periodic Function, Line Integral, Periodic Points of Complex Quadratic Mappings, Mellin Inversion Theorem, Schramm-loewner Evolution, Bieberbach Conjecture, Complex Quadratic Polynomial, Kramers-kronig Relation, Radius of Convergence, Algebraic Geometry and Analytic Geometry, Cayley Transform, Argument, Partial Fractions in Complex Analysis, Riemann-hilbert Problem, Conformal Pictures, Motor Variable, Subharmonic Function, Liouville's Theorem, Edge-Of-The-Wedge Theorem, Monodromy Theorem, Regular Singular Point, Lacunary Function, Paley-wiener Theorem, Schwarz Lemma, Weierstrass Factorization Theorem, Conformal Radius, Lee-yang Theorem, Euler's Continued Fraction Formula, Logarithmic Derivative, Rouché's Theorem, Antiderivative, Cousin Problems, Filled Julia Set, Picard Theorem, Several Complex Variables, Morera's Theorem, Nachbin's Theorem, Ramification, Schwarzian Derivative, Analytic Capacity, Domain Coloring, Siegel Disc, Analyticity of Holomorphic Functions, Fatou's Theorem, Bombieri Norm, Hyperfunction, Complex Conjugate Root Theorem, Generalised Circle, Principal Value, Doubly-Periodic Function, Jordan's Lemma, Normal Family, Nörlund-rice Integral, Positive-Defi...More: http://booksllc.net/?id=5826 ... Read more

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This is a reproduction of a book published before 1923.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process.We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide.We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

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Publisher: Boston : GinnPublication date: 1889Subjects: AlgebraNotes: This is an OCR reprint. There may be typos or missing text. There are no illustrations or indexes.When you buy the General Books edition of this book you get free trial access to Million-Books.com where you can select from more than a million books for free. You can also preview the book there. ... Read more

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This digital document is a journal article from European Journal of Operational Research, published by Elsevier in 2007. The article is delivered in HTML format and is available in your Amazon.com Media Library immediately after purchase. You can view it with any web browser.

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The aim of this paper is to present a novel, transparent approach to a well-established field: the deep methods and applications of the complete analysis of continuous optimization problems. Standard descents give a unified approach to all standard necessary conditions, including the Lagrange multiplier rule, the Karush-Kuhn-Tucker conditions and the second order conditions. Nonstandard descents lead to new necessary conditions. These can be used to give surprising proofs of deep central results of fields that are generally viewed as distinct from optimization: the fundamental theorem of algebra, the maximum and the minimum principle of complex function theory, the separation theorems for convex sets, the orthogonal diagonalization of symmetric matrices and the implicit function theorem. These optimization proofs compare favorably with the usual proofs and are all based on the same strategy. This paper is addressed to all practitioners of optimization methods from many fields who are interested in fully understanding the foundations of these methods and of the central results above. ... Read more

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