Previous edition sold 2000 copies in 3 years; Explores the subtle connections between Number Theory, Classical Geometry and Modern Algebra; Over 180 illustrations, as well as text and Maple files, are available via the web facilitate understanding: http://mathsgi01.rutgers.edu/cgi-bin/wrap/gtoth/; Contains an insert with 4-color illustrations; Includes numerous examples and worked-out problems
Author(s): Gabor Toth
Series: Undergraduate Texts in Mathematics
Edition: 2nd
Publisher: Springer
Year: 2002
Language: English
Pages: 473
Contents......Page 20
Preface to the Second Edition......Page 8
Preface to the First Edition......Page 12
Acknowledgments......Page 18
Section 1 "A Number Is a Multitude Composed of Units"—Euclid......Page 24
Web Sites......Page 29
Section 2 "… There Are No Irrational Numbers at All"—Kronecker......Page 30
Problems......Page 44
Web Sites......Page 48
Section 3 Rationality, Elliptic Curves, and Fermat's Last Theorem......Page 49
Problems......Page 75
Web Sites......Page 77
Section 4 Algebraic or Transcendental?......Page 78
Problems......Page 83
Section 5 Complex Arithmetic......Page 85
Problems......Page 94
Section 6 Quadratic, Cubic, and Quartic Equations......Page 95
Problems......Page 103
Section 7 Stereographic Projection......Page 106
Problems......Page 111
Web Site......Page 112
Section 8 Proof of the Fundamental Theorem of Algebra......Page 113
Problems......Page 116
Web Site......Page 118
Section 9 Symmetries of Regular Polygons......Page 119
Problems......Page 128
Web Sites......Page 129
Section 10 Discrete Subgroups of Iso (R[sup(2)])......Page 130
Problems......Page 143
Web Sites......Page 144
Section 11 Möbius Geometry......Page 145
Problems......Page 153
Section 12 Complex Linear Fractional Transformations......Page 154
Problems......Page 160
Section 13 "Out of Nothing I Have Created a New Universe"—Bolyai......Page 162
Problems......Page 179
Section 14 Fuchsian Groups......Page 181
Problems......Page 194
Section 15 Riemann Surfaces......Page 196
Problems......Page 220
Web Site......Page 221
Section 16 General Surfaces......Page 222
Web Site......Page 231
Section 17 The Five Platonic Solids......Page 232
Problems......Page 271
Film......Page 277
Section 18 Finite Möbius Groups......Page 278
Section 19 Detour in Topology: Euler–Poincaré Characteristic......Page 289
Film......Page 301
Section 20 Detour in Graph Theory: Euler, Hamilton, and the Four Color Theorem......Page 302
Problems......Page 317
Web Sites......Page 320
Section 21 Dimension Leap......Page 321
Problems......Page 327
Section 22 Quaternions......Page 328
Problems......Page 338
Web Sites......Page 339
Section 23 Back to R[sup(3)]!......Page 340
Problems......Page 351
Section 24 Invariants......Page 352
Problem......Page 367
Section 25 The Icosahedron and the Unsolvable Quintic......Page 368
A. Polyhedral Equations......Page 369
B. Hypergeometric Functions......Page 371
C. The Tschirnhaus Transformation......Page 374
D. Quintic Resolvents of the Icosahedral Equation......Page 378
E. Solvability of the Quintic à la Klein......Page 386
F. Geometry of the Canonical Equation: General Considerations......Page 388
G. Geometry of the Canonical Equation: Explicit Formulas......Page 392
Problems......Page 400
Section 26 The Fourth Dimension......Page 403
Problems......Page 417
Film......Page 418
Appendix A Sets......Page 420
Appendix B Groups......Page 422
Appendix C Topology......Page 426
Appendix D Smooth Maps......Page 430
Appendix E The Hypergeometric Differential Equation and the Schwarzian......Page 432
Appendix F Galois Theory......Page 442
Solutions for 100 Selected Problems......Page 448
B......Page 466
D......Page 467
G......Page 468
I......Page 469
N......Page 470
Q......Page 471
T......Page 472
Z......Page 473
