The Fundamental Theorem Of Algebra

Author : Benjamin Fine,Gerhard Rosenberger
Publisher : Springer Science & Business Media
Page : 220 pages
File Size : 52,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461219286

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The Fundamental Theorem of Algebra by Benjamin Fine,Gerhard Rosenberger Pdf

The 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.

Author : Benjamin Fine,Gerhard Rosenberger
Publisher : Springer Science & Business Media
Page : 232 pages
File Size : 55,7 Mb
Release : 1997-06-20
Category : Mathematics
ISBN : 0387946578

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The Fundamental Theorem of Algebra by Benjamin Fine,Gerhard Rosenberger Pdf

The 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.

Author : Isaiah Lankham,Bruno Nachtergaele,Anne Schilling
Publisher : World Scientific Publishing Company
Page : 208 pages
File Size : 41,7 Mb
Release : 2015-11-30
Category : Mathematics
ISBN : 9789814723794

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Linear Algebra as an Introduction to Abstract Mathematics by Isaiah Lankham,Bruno Nachtergaele,Anne Schilling Pdf

This is an introductory textbook designed for undergraduate mathematics majors with an emphasis on abstraction and in particular, the concept of proofs in the setting of linear algebra. Typically such a student would have taken calculus, though the only prerequisite is suitable mathematical grounding. The purpose of this book is to bridge the gap between the more conceptual and computational oriented undergraduate classes to the more abstract oriented classes. The book begins with systems of linear equations and complex numbers, then relates these to the abstract notion of linear maps on finite-dimensional vector spaces, and covers diagonalization, eigenspaces, determinants, and the Spectral Theorem. Each chapter concludes with both proof-writing and computational exercises.

Author : Allan Clark
Publisher : Courier Corporation
Page : 224 pages
File Size : 48,5 Mb
Release : 2012-07-06
Category : Mathematics
ISBN : 9780486140353

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Elements of Abstract Algebra by Allan Clark Pdf

Lucid coverage of the major theories of abstract algebra, with helpful illustrations and exercises included throughout. Unabridged, corrected republication of the work originally published 1971. Bibliography. Index. Includes 24 tables and figures.

Author : Hermann Weyl
Publisher : Princeton University Press
Page : 244 pages
File Size : 43,9 Mb
Release : 1998
Category : Mathematics
ISBN : 0691059179

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Algebraic Theory of Numbers by Hermann Weyl Pdf

This work explores the fundamental concepts in arithmetic. It begins with the definitions and properties of algebraic fields. The theory of divisibility is then discussed. There follows an introduction to p-adic numbers and then culminates with an extensive examination of algebraic number fields.

Author : Harry Pollard,Harold G. Diamond
Publisher : American Mathematical Soc.
Page : 162 pages
File Size : 48,7 Mb
Release : 1975-12-31
Category : Algebraic number theory
ISBN : 9781614440093

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The Theory of Algebraic Numbers: Second Edition by Harry Pollard,Harold G. Diamond Pdf

This monograph makes available, in English, the elementary parts of classical algebraic number theory. This second edition follows closely the plan and style of the first edition. The principal changes are the correction of misprints, the expansion or simplification of some arguments, and the omission of the final chapter on units in order to make way for the introduction of some two hundred problems.

Author : Peter Borwein,Tamas Erdelyi
Publisher : Springer Science & Business Media
Page : 491 pages
File Size : 42,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461207931

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Polynomials and Polynomial Inequalities by Peter Borwein,Tamas Erdelyi Pdf

After an introduction to the geometry of polynomials and a discussion of refinements of the Fundamental Theorem of Algebra, the book turns to a consideration of various special polynomials. Chebyshev and Descartes systems are then introduced, and Müntz systems and rational systems are examined in detail. Subsequent chapters discuss denseness questions and the inequalities satisfied by polynomials and rational functions. Appendices on algorithms and computational concerns, on the interpolation theorem, and on orthogonality and irrationality round off the text. The book is self-contained and assumes at most a senior-undergraduate familiarity with real and complex analysis.

Author : Benjamin Fine,Gerhard Rosenberger
Publisher : Unknown
Page : 228 pages
File Size : 51,6 Mb
Release : 1997-06-20
Category : Electronic
ISBN : 1461219299

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The Fundamental Theorem of Algebra by Benjamin Fine,Gerhard Rosenberger Pdf

Author : Charles C Pinter
Publisher : Courier Corporation
Page : 402 pages
File Size : 55,6 Mb
Release : 2010-01-14
Category : Mathematics
ISBN : 9780486474175

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A Book of Abstract Algebra by Charles C Pinter Pdf

Accessible but rigorous, this outstanding text encompasses all of the topics covered by a typical course in elementary abstract algebra. Its easy-to-read treatment offers an intuitive approach, featuring informal discussions followed by thematically arranged exercises. This second edition features additional exercises to improve student familiarity with applications. 1990 edition.

Author : Benjamin Fine,Anthony M. Gaglione,Gerhard Rosenberger
Publisher : JHU Press
Page : 583 pages
File Size : 47,5 Mb
Release : 2014-07-01
Category : Mathematics
ISBN : 9781421411774

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Introduction to Abstract Algebra by Benjamin Fine,Anthony M. Gaglione,Gerhard Rosenberger Pdf

A new approach to abstract algebra that eases student anxieties by building on fundamentals. Introduction to Abstract Algebra presents a breakthrough approach to teaching one of math's most intimidating concepts. Avoiding the pitfalls common in the standard textbooks, Benjamin Fine, Anthony M. Gaglione, and Gerhard Rosenberger set a pace that allows beginner-level students to follow the progression from familiar topics such as rings, numbers, and groups to more difficult concepts. Classroom tested and revised until students achieved consistent, positive results, this textbook is designed to keep students focused as they learn complex topics. Fine, Gaglione, and Rosenberger's clear explanations prevent students from getting lost as they move deeper and deeper into areas such as abelian groups, fields, and Galois theory. This textbook will help bring about the day when abstract algebra no longer creates intense anxiety but instead challenges students to fully grasp the meaning and power of the approach. Topics covered include: • Rings • Integral domains • The fundamental theorem of arithmetic • Fields • Groups • Lagrange's theorem • Isomorphism theorems for groups • Fundamental theorem of finite abelian groups • The simplicity of An for n5 • Sylow theorems • The Jordan-Hölder theorem • Ring isomorphism theorems • Euclidean domains • Principal ideal domains • The fundamental theorem of algebra • Vector spaces • Algebras • Field extensions: algebraic and transcendental • The fundamental theorem of Galois theory • The insolvability of the quintic

Author : Jean-Pierre Tignol
Publisher : World Scientific Publishing Company
Page : 324 pages
File Size : 43,8 Mb
Release : 2015-12-28
Category : Mathematics
ISBN : 9789814704717

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Galois' Theory of Algebraic Equations by Jean-Pierre Tignol Pdf

The book gives a detailed account of the development of the theory of algebraic equations, from its origins in ancient times to its completion by Galois in the nineteenth century. The appropriate parts of works by Cardano, Lagrange, Vandermonde, Gauss, Abel, and Galois are reviewed and placed in their historical perspective, with the aim of conveying to the reader a sense of the way in which the theory of algebraic equations has evolved and has led to such basic mathematical notions as "group" and "field". A brief discussion of the fundamental theorems of modern Galois theory and complete proofs of the quoted results are provided, and the material is organized in such a way that the more technical details can be skipped by readers who are interested primarily in a broad survey of the theory. In this second edition, the exposition has been improved throughout and the chapter on Galois has been entirely rewritten to better reflect Galois' highly innovative contributions. The text now follows more closely Galois' memoir, resorting as sparsely as possible to anachronistic modern notions such as field extensions. The emerging picture is a surprisingly elementary approach to the solvability of equations by radicals, and yet is unexpectedly close to some of the most recent methods of Galois theory.

Author : Chee-Keng Yap
Publisher : Oxford University Press on Demand
Page : 511 pages
File Size : 44,6 Mb
Release : 2000
Category : Computers
ISBN : 0195125169

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Fundamental Problems of Algorithmic Algebra by Chee-Keng Yap Pdf

Popular computer algebra systems such as Maple, Macsyma, Mathematica, and REDUCE are now basic tools on most computers. Efficient algorithms for various algebraic operations underlie all these systems. Computer algebra, or algorithmic algebra, studies these algorithms and their properties and represents a rich intersection of theoretical computer science with classical mathematics. Fundamental Problems of Algorithmic Algebra provides a systematic and focused treatment of a collection of core problemsthe computational equivalents of the classical Fundamental Problem of Algebra and its derivatives. Topics covered include the GCD, subresultants, modular techniques, the fundamental theorem of algebra, roots of polynomials, Sturm theory, Gaussian lattice reduction, lattices and polynomial factorization, linear systems, elimination theory, Grobner bases, and more. Features · Presents algorithmic ideas in pseudo-code based on mathematical concepts and can be used with any computer mathematics system · Emphasizes the algorithmic aspects of problems without sacrificing mathematical rigor · Aims to be self-contained in its mathematical development · Ideal for a first course in algorithmic or computer algebra for advanced undergraduates or beginning graduate students

Author : Thomas W. Hungerford
Publisher : Springer Science & Business Media
Page : 523 pages
File Size : 47,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461261018

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Algebra by Thomas W. Hungerford Pdf

Finally a self-contained, one volume, graduate-level algebra text that is readable by the average graduate student and flexible enough to accommodate a wide variety of instructors and course contents. The guiding principle throughout is that the material should be presented as general as possible, consistent with good pedagogy. Therefore it stresses clarity rather than brevity and contains an extraordinarily large number of illustrative exercises.

Author : H. P. F. Swinnerton-Dyer
Publisher : Cambridge University Press
Page : 164 pages
File Size : 45,9 Mb
Release : 2001-02-22
Category : Mathematics
ISBN : 0521004233

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A Brief Guide to Algebraic Number Theory by H. P. F. Swinnerton-Dyer Pdf

Broad graduate-level account of Algebraic Number Theory, first published in 2001, including exercises, by a world-renowned author.

Author : Ian Stewart,David Tall
Publisher : CRC Press
Page : 334 pages
File Size : 52,8 Mb
Release : 2001-12-12
Category : Mathematics
ISBN : 9781439864081

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Algebraic Number Theory and Fermat's Last Theorem by Ian Stewart,David Tall Pdf

First published in 1979 and written by two distinguished mathematicians with a special gift for exposition, this book is now available in a completely revised third edition. It reflects the exciting developments in number theory during the past two decades that culminated in the proof of Fermat's Last Theorem. Intended as a upper level textbook, it