Abstract Algebra and Famous Impossibilities: Squaring the Circle, Doubling the Cube, Trisecting an Angle, and Solving Quintic Equations: Undergraduate Texts in Mathematics
Autor Sidney A. Morris, Arthur Jones, Kenneth R. Pearsonen Limba Engleză Paperback – 28 noi 2023
Beginning with a brief account of the history of these fabled problems, the book goes on to present the theory of fields, polynomials, field extensions, and irreducible polynomials. Straightedge and compass constructions establish the standards for constructability, and offer a glimpse into why squaring, doubling, and trisecting appeared so tractable to professional and amateur mathematicians alike. However, the connection between geometry and algebra allows the reader to bypass two millennia of failed geometric attempts, arriving at the elegant algebraic conclusion that such constructions are impossible. From here, focus turns to a challenging problem within algebra itself: finding a general formula for solving a quintic polynomial. The proof of the impossibility of this task is presented using Abel’s original approach.
Abstract Algebra and Famous Impossibilities illustrates the enormous power of algebraic abstraction by exploring several notable historical triumphs. This new edition adds the fourth impossibility: solving general quintic equations. Students and instructors alike will appreciate the illuminating examples, conversational commentary, and engaging exercises that accompany each section. A first course in linear algebra is assumed, along with a basic familiarity with integral calculus.
Toate formatele și edițiile | Preț | Express |
---|---|---|
Paperback (1) | 417.26 lei 38-45 zile | |
Springer International Publishing – 28 noi 2023 | 417.26 lei 38-45 zile | |
Hardback (1) | 434.93 lei 38-45 zile | |
Springer International Publishing – 28 noi 2022 | 434.93 lei 38-45 zile |
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Specificații
ISBN-13: 9783031057007
ISBN-10: 3031057007
Pagini: 218
Ilustrații: XXII, 218 p. 29 illus.
Dimensiuni: 155 x 235 mm
Ediția:2nd ed. 2022
Editura: Springer International Publishing
Colecția Springer
Seriile Undergraduate Texts in Mathematics, Readings in Mathematics
Locul publicării:Cham, Switzerland
ISBN-10: 3031057007
Pagini: 218
Ilustrații: XXII, 218 p. 29 illus.
Dimensiuni: 155 x 235 mm
Ediția:2nd ed. 2022
Editura: Springer International Publishing
Colecția Springer
Seriile Undergraduate Texts in Mathematics, Readings in Mathematics
Locul publicării:Cham, Switzerland
Cuprins
1. Algebraic Preliminaries.- 2. Algebraic Numbers and Their Polynomials.- 3. Extending Fields.- 4. Irreducible Polynomials.- 5. Straightedge and Compass Constructions.- 6. Proofs of the Geometric Impossibilities.- 7. Zeros of Polynomials of Degrees 2, 3, and 4.- 8. Quintic Equations 1: Symmetric Polynomials.- 9. Quintic Equations II: The Abel–Ruffini Theorem.- 10. Transcendence of e and π.- 11. An Algebraic Postscript.- 12. Other Impossibilities: Regular Polygons and Integration in Finite Terms.- References.- Index.
Notă biografică
Sidney A. Morris is Emeritus Professor at the Federation University, Australia (formerly University of Ballarat) and Adjunct Professor at La Trobe University, Australia. His primary research is in topological groups, topology, and transcendental number theory, with broader interests including early detection of muscle wasting diseases, health informatics, and predicting the Australian stock exchange. He is the author of several books.
Arthur Jones [1934–2006] and Kenneth R. Pearson [1943–2015] were Professors in Mathematics at La Trobe University, Australia. Each had a great passion for teaching and for mathematics.
Arthur Jones [1934–2006] and Kenneth R. Pearson [1943–2015] were Professors in Mathematics at La Trobe University, Australia. Each had a great passion for teaching and for mathematics.
Textul de pe ultima copertă
This textbook develops the abstract algebra necessary to prove the impossibility of four famous mathematical feats: squaring the circle, trisecting the angle, doubling the cube, and solving quintic equations. All the relevant concepts about fields are introduced concretely, with the geometrical questions providing motivation for the algebraic concepts. By focusing on problems that are as easy to approach as they were fiendishly difficult to resolve, the authors provide a uniquely accessible introduction to the power of abstraction.
Beginning with a brief account of the history of these fabled problems, the book goes on to present the theory of fields, polynomials, field extensions, and irreducible polynomials. Straightedge and compass constructions establish the standards for constructability, and offer a glimpse into why squaring, doubling, and trisecting appeared so tractable to professional and amateur mathematicians alike. However, the connection between geometry and algebra allows the reader to bypass two millennia of failed geometric attempts, arriving at the elegant algebraic conclusion that such constructions are impossible. From here, focus turns to a challenging problem within algebra itself: finding a general formula for solving a quintic polynomial. The proof of the impossibility of this task is presented using Abel’s original approach.
Abstract Algebra and Famous Impossibilities illustrates the enormous power of algebraic abstraction by exploring several notable historical triumphs. This new edition adds the fourth impossibility: solving general quintic equations. Students and instructors alike will appreciate the illuminating examples, conversational commentary, and engaging exercises that accompany each section. A first course in linear algebra is assumed, along with a basic familiarity with integral calculus.
Beginning with a brief account of the history of these fabled problems, the book goes on to present the theory of fields, polynomials, field extensions, and irreducible polynomials. Straightedge and compass constructions establish the standards for constructability, and offer a glimpse into why squaring, doubling, and trisecting appeared so tractable to professional and amateur mathematicians alike. However, the connection between geometry and algebra allows the reader to bypass two millennia of failed geometric attempts, arriving at the elegant algebraic conclusion that such constructions are impossible. From here, focus turns to a challenging problem within algebra itself: finding a general formula for solving a quintic polynomial. The proof of the impossibility of this task is presented using Abel’s original approach.
Abstract Algebra and Famous Impossibilities illustrates the enormous power of algebraic abstraction by exploring several notable historical triumphs. This new edition adds the fourth impossibility: solving general quintic equations. Students and instructors alike will appreciate the illuminating examples, conversational commentary, and engaging exercises that accompany each section. A first course in linear algebra is assumed, along with a basic familiarity with integral calculus.
Caracteristici
Motivates the development of algebraic concepts through tantalizing geometric questions from history Illustrates the power of algebraic abstraction for tackling concrete questions Engages the reader with abundant examples, commentary, and exercises