Abstract Algebra and Famous Impossibilities / Edition 1

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The famous problems of squaring the circle, doubling the cube, and trisecting the angle have captured the imagination of both professional and amateur mathematician for over two thousand years. These problems, however, have not yielded to purely geometrical methods. It was only the development of abstract algebra in the nineteenth century which enabled mathematicians to arrive at the surprising conclusion that these constructions are not possible. This text aims to develop the abstract algebra.

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Editorial Reviews

Develops the abstract algebra necessary to prove the impossibility of constructions to square the circle, double the cube, trisect the angle by geometrical means. Assumes linear algebra. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

  • ISBN-13: 9780387976617
  • Publisher: Springer New York
  • Publication date: 9/24/1991
  • Series: Universitext Series
  • Edition description: 1st ed. 1991. Corr. 2nd printing 1993
  • Edition number: 1
  • Pages: 189
  • Product dimensions: 0.43 (w) x 6.14 (h) x 9.21 (d)

Table of Contents

0.1 Three Famous Problems.- 0.2 Straightedge and Compass Constructions.- 0.3 Impossibility of the Constructions.- 1 Algebraic Preliminaries.- 1.1 Fields, Rings and Vector Spaces.- 1.2 Polynomials.- 1.3 The Division Algorithm.- 1.4 The Rational Roots Test.- Appendix to Chapter 1.- 2 Algebraic Numbers and Their Polynomials.- 2.1 Algebraic Numbers.- 2.2 Monic Polynomials.- 2.3 Monic Polynomials of Least Degree.- 3 Extending Fields.- 3.1 An Illustration: $$\mathbb{Q}(\sqrt 2 )$$.- 3.2 Construction of $$\mathbb{F}(\alpha )$$.- 3.3 Iterating the Construction.- 3.4 Towers of Fields.- 4 Irreducible Polynomials.- 4.1 Irreducible Polynomials.- 4.2 Reducible Polynomials and Zeros.- 4.3 Irreducibility and irr$$(\alpha ,\mathbb{F})$$.- 4.4 Finite-dimensional Extensions.- 5 Straightedge and Compass Constructions.- 5.1 Standard Straightedge and Compass Constructions.- 5.2 Products, Quotients, Square Roots.- 5.3 Rules for Straightedge and Compass Constructions.- 5.4 Constructible Numbers and Fields.- 6 Proofs of the Impossibilities.- 6.1 Non-Constructible Numbers.- 6.2 The Three Constructions are Impossible.- 6.3 Proving the “All Constructibles Come From Square Roots” Theorem.- 7 Transcendence of e and—.- 7.1 Preliminaries.- 7.2 e is Transcendental.- 7.3 Preliminaries on Symmetric Polynomials.- 7.4— is Transcendental — Part 1.- 7.5 Preliminaries on Complex-valued Integrals.- 7.6— is Transcendental — Part 2.- 8 An Algebraic Postscript.- 8.1 The Ring $$\mathbb{F}\left[ X \right]_{p(X)}$$.- 8.2 Division and Reciprocals in $$\mathbb{F}\left[ X \right]_{p(X)}$$.- 8.3 Reciprocals in $$\mathbb{F}\left( \alpha \right)$$.- 9 Other Impossibilities and Abstract Algebra.- 9.1 Construction of Regular Polygons.- 9.2 Solution of Quintic Equations.- 9.3 Integration in Closed Form.

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