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E-grāmata: Abstract Algebra and Famous Impossibilities: Squaring the Circle, Doubling the Cube, Trisecting an Angle, and Solving Quintic Equations

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  • Formāts: EPUB+DRM
  • Sērija : Readings in Mathematics
  • Izdošanas datums: 26-Nov-2022
  • Izdevniecība: Springer International Publishing AG
  • Valoda: eng
  • ISBN-13: 9783031056987
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Abstract Algebra and Famous Impossibilities: Squaring the Circle, Doubling the Cube, Trisecting an Angle, and Solving Quintic EquationsPalielināt
  • Formāts: EPUB+DRM
  • Sērija : Readings in Mathematics
  • Izdošanas datums: 26-Nov-2022
  • Izdevniecība: Springer International Publishing AG
  • Valoda: eng
  • ISBN-13: 9783031056987
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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 Abels 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.
1 Algebraic Preliminaries
1(18)
1.1 Fields, Rings and Vector Spaces
1(5)
Exercises 1.1
5(1)
1.2 Polynomials
6(4)
Exercises 1.2
9(1)
1.3 The Division Algorithm
10(2)
Exercises 1.3
11(1)
1.4 The Rational Roots Test
12(7)
Exercises 1.4
14(1)
Appendix to
Chapter 1
15(2)
Additional Reading for
Chapter 1
17(2)
2 Algebraic Numbers and Their Polynomials
19(12)
2.1 Algebraic Numbers
20(4)
Exercises 2.1
22(2)
2.2 Monic Polynomials
24(1)
Exercises 2.2
25(1)
2.3 Monic Polynomials of Least Degree
25(6)
Exercises 2.3
28(2)
Additional Reading for
Chapter 2
30(1)
3 Extending Fields
31(20)
3.1 An Illustration: Q(2)
31(4)
Exercises 3.1
34(1)
3.2 Construction of F(α)
35(6)
Exercises 3.2
40(1)
3.3 Iterating the Construction
41(2)
Exercises 3.3
42(1)
3.4 Towers of Fields
43(8)
Exercises 3.4
46(3)
Additional Reading for
Chapter 3
49(2)
4 Irreducible Polynomials
51(14)
4.1 Irreducible Polynomials
51(2)
Exercises 4.1
53(1)
4.2 Reducible Polynomials and Zeros
53(4)
Exercises 4.2
56(1)
4.3 Irreducibility and irr(α, F)
57(3)
Exercises 4.3
59(1)
4.4 Finite-dimensional Extensions
60(5)
Exercises 4.4
61(1)
Additional Reading for
Chapter 4
62(3)
5 Straightedge and Compass Constructions
65(24)
5.1 Standard Straightedge and Compass Constructions
65(11)
Exercises 5.1
72(4)
5.2 Products, Quotients, Square Roots
76(3)
Exercises 5.2
78(1)
5.3 Rules for Straightedge and Compass Constructions
79(5)
Exercises 5.3
84(1)
5.4 Constructible Numbers and Fields
84(5)
Exercises 5.4
87(1)
Additional Reading for
Chapter 5
87(2)
6 Proofs of the Geometric Impossibilities
89(12)
6.1 Non-Constructible Numbers
89(4)
Exercises 6.1
92(1)
6.2 The Three Geometric Constructions are Impossible
93(3)
Exercises 6.2
94(2)
6.3 All Constructibles Come From Square Roots" Theorem
96(5)
Exercises 6.3
99(1)
Additional Reading for
Chapter 6
100(1)
7 Zeros of Polynomials of Degrees 2, 3, and 4
101(10)
7.1 Solving Quadratic Equations
102(1)
Exercises 7.1
103(1)
7.2 Solving Cubic Equations
103(4)
Exercises 7.2
106(1)
7.3 Solving Quarlic Equations
107(4)
Exercises 7.3
108(1)
Additional Reading for
Chapter 7
109(2)
8 Quintic Equations I: Symmetric Polynomials
111(18)
8.1 Brief History of the Quintic Equation: 1683--1826
111(2)
8.2 Fundamental Theorem of Algebra
113(3)
Exercises 8.2
114(2)
8.3 Primitive and Symmetric Polynomials
116(13)
Exercises 8.3
127(1)
Additional Reading for
Chapter 8
128(1)
9 Quintic Equations II: The Abel--Ruffini Theorem
129(16)
9.1 Algebraically Soluble Polynomials
129(2)
Exercises 9.1
131(1)
9.2 The Number of Real Number Zeros of an Irreducible Polynomial
131(6)
Exercises 9.2
137(1)
9.3 Kronecker's Theorem and the Abel--Rumni Theorem
137(8)
Exercises 9.3
143(1)
Additional Reading for
Chapter 9
144(1)
10 Transcendence of e and π
145(36)
10.1 Preliminaries
145(8)
Exercises 10.1
152(1)
10.2 E is Transcendental
153(9)
Exercises 10.2
160(2)
10.3 π is Transcendental -- Part 1
162(3)
Exercises 10.3
165(1)
10.4 Preliminaries on Complex-valued Integrals
165(3)
Exercises 10.4
168(1)
10.5 π is Transcendental -- Part 2
168(7)
Exercises 10.5
174(1)
10.6 Transcendental Number Theory
175(6)
Exercises 10.6
178(1)
Additional Reading for
Chapter 10
179(2)
11 An Algebraic Postscript
181(12)
11.1 The Ring F[ X]p(x)
181(2)
Exercises 11.1
182(1)
11.2 Division and Reciprocals in F[ X]p(x)
183(4)
Exercises 11.2
187(1)
11.3 Reciprocals in F(α)
187(6)
Exercises 11.3
190(1)
Additional Reading for
Chapter 11
191(2)
12 Other Impossibilities: Regular Polygons and Integration in Finite Terms
193(14)
12.1 Construction of Regular Polygons
193(1)
12.2 Integration in Closed Form
194(13)
Exercises 12.2
198(1)
Additional Reading for
Chapter 12
198(3)
References
201(6)
Index 207
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 [ 19342006] and Kenneth R. Pearson [ 19432015] were Professors in Mathematics at La Trobe University, Australia. Each had a great passion for teaching and for mathematics.
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