Glimpses of Algebra and Geometry (Undergraduate Texts in Mathematics)

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. The text is divided into subtexts of four levels (indicated by the card symbols in Bridge) according to the readers aptitude. 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. Highlights of GLIMPSES include dicussions of: Rationality, Elliptic Curves and Fermat's Last Theorem, Fundamental Theorem of Algebra, Mbuis Geometry, Hyperbolic Geometry and Riemann Surfaces, Platonic Solids, Topology of Surfaces, The Four Color Theorem and The Fourth Dimension. Material from this volume can be taught the traditional way using slides, or interactively in a computer lab or teaching facility equipped with a PC or a workstation connected to an LCD-panel. Because there is a great deal of information, interactive graphics, and animations available over the Web, the author also includes recommended Web sites at the end of each section.

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 ... will benefit from this carefully worked-out textbook." (J. Lang, Internationale Mathematische Nachrichten, Vol. 57 (192), 2003)