Glimpses of Algebra and Geometry

1. Front Matter

   Pages i-xxii

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2. “A Number Is a Multitude Composed of Units”—Euclid

   Pages 1-6

3. “... There Are No Irrational Numbers at All”—Kronecker

   Pages 7-25

4. Rationality, Elliptic Curves, and Fermat’s Last Theorem

   Pages 26-54

5. Algebraic or Transcendental?

   Pages 55-61

6. Complex Arithmetic

   Pages 62-71

7. Quadratic, Cubic, and Quartic Equations

   Pages 72-82

8. Stereographic Projection

   Pages 83-89

9. Proof of the Fundamental Theorem of Algebra

   Pages 90-95

10. Symmetries of Regular Polygons

    Pages 96-106

11. Discrete Subgroups of Iso (R2)

    Pages 107-121

12. Möbius Geometry

    Pages 122-130

13. Complex Linear Fractional Transformations

    Pages 131-138

14. “Out of Nothing I Have Created a New Universe”—Bolyai

    Pages 139-157

15. Fuchsian Groups

    Pages 158-172

16. Riemann Surfaces

    Pages 173-198

17. General Surfaces

    Pages 199-208

18. The Five Platonic Solids

    Pages 209-254

19. Finite Möbius Groups

    Pages 255-265

20. Detour in Topology: Euler-Poincaré Characteristic

    Pages 266-278

The purpose of Glimpses of Algebra and Geometry is to fill a gap between undergraduate and graduate mathematics studies. It is one of the few undergraduate texts to explore the subtle and sometimes puzzling connections between Number Theory, Classical Geometry and Modern Algebra in a clear and easily understandable style. Over 160 computer-generated images, accessible to readers via the World Wide Web, facilitate an understanding of mathematical concepts and proofs even further.
Glimpses also sheds light on some of the links between the first recorded intellectual attempts to solve ancient problems of Number Theory and Geometry and twentieth century mathematics. GLIMPSES will appeal to students who wish to learn modern mathematics, but have few prerequisite courses, and to high-school teachers who always had a keen interest in mathematics, but seldom the time to pursue background technicalities. Even postgraduate mathematicians will enjoy being able to browse through a number of mathematical disciplines in one sitting.
This new edition includes invaluable improvements throughout the text, including an in-depth treatment of root formulas, a detailed and complete classification of finite Möbius groups a la Klein, and a quick, direct, and modern approach to Felix Kleins "Normalformsatz," the main result of his spectacular theory of icosahedron and his solution of the irreducible quintic in terms of hypergeometric functions.
Gabor Toth is the Chair and Graduate Director of the Department of Mathematical Sciences at Rutgers University, Camden. His previous publications include Finite Mobius Groups, Spherical Minimal Immersions and Moduli (2001), Harmonic Maps and Minimal Immersion Through Representation Theory (1990) and Harmonic and Minimal Maps with Applications in Geometry and Physics (1984). Professor Toths main fields of interest involve the geometry of eigenmaps and spherical minimal immersions and the visualization of mathematics viacomputers.

• Hypergeometric function
• Riemann surface
• algebra
• elliptic curve
• number theory
• transformation

From the reviews of the second edition:

"Toth’s ‘Glimpses’ offer selected material that connect algebra and geometry … . This second edition is a revised and substantially expanded version, so for example it includes a detailed treatment of the solution of the cubic and quartic, as well as a long new chapter on Klein’s famous work on the quintic and the icosahedron." (Günter M. Ziegler, Zentralblatt MATH, Vol. 1027, 2004)

"The book is intended – and really manages it – to fill undergraduates with enthusiasm to reach the graduate level. … the author presents various topics of number theory, geometry and algebra and at the same time shows their connection resp. interplay, thus making the study lively and fascinating for the reader. … information on advanced websites and films show how carefully the author has done his job. So this second edition hopefully will not be the last one." (G. Kowol, Monatshefte für Mathematik, Vol. 141 (2), 2004)

"The text covers a wide range of topics and gives a taste of advanced material in number theory, geometry and algebra, particularly where these fields overlap. … there are plenty of references for the interested reader who wishes to pursue a particular topic in greater depth. … the accessibility of the format and the flow of the material combine to create an entertaining and informative work. I recommend the text as a good read for mathematicians of all specialities. " (Stephen Lucas, The Australian Mathematical Society Gazette, Vol. 30 (4), 2003)

"This is the second, much revised and augmented edition of the book originally published in 1998. It intends to close the gap between undergraduate and graduate studies in number theory, classical geometry and modern algebra. … Each of the chapters is a good read and the book adds up to a wholly appealing entity. … It can be warmly recommended … . I can well imagine that teachers … as well as scientists … willbenefit from this carefully worked-out textbook." (J. Lang, Internationale Mathematische Nachrichten, Vol. 57 (192), 2003)

• Department of Mathematical Sciences, Rutgers University, Camden, USA

  Gabor Toth

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