Overview

"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 besides innumerable improvements throughout the text an in-depth treatment of root formulas, a detailed and complete classification of finite Mobius groups a la Klein, and a quick, direct, and modern approach to Felix Klein's ""Normalformsatz,"" the main result of his spectacular theory of the icosahedron and his solution of the irreducible quintic in terms of hypergeometric functions."

Full Product Details

Author:   Gabor Toth
Publisher:   Springer-Verlag New York Inc.
Imprint:   Springer-Verlag New York Inc.
Edition:   2nd ed. 2002
Dimensions:   Width: 17.80cm , Height: 3.30cm , Length: 23.50cm
Weight:   1.112kg
ISBN:  

9780387953458

ISBN 10:   0387953450
Pages:   450
Publication Date:   02 May 2002
Audience:   College/higher education ,  Professional and scholarly ,  Undergraduate ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

“A Number Is a Multitude Composed of Units”—Euclid.- “... There Are No Irrational Numbers at All”—Kronecker.- Rationality, Elliptic Curves, and Fermat’s Last Theorem.- Algebraic or Transcendental?.- Complex Arithmetic.- Quadratic, Cubic, and Quartic Equations.- Stereographic Projection.- Proof of the Fundamental Theorem of Algebra.- Symmetries of Regular Polygons.- Discrete Subgroups of Iso (R2).- Möbius Geometry.- Complex Linear Fractional Transformations.- “Out of Nothing I Have Created a New Universe”—Bolyai.- Fuchsian Groups.- Riemann Surfaces.- General Surfaces.- The Five Platonic Solids.- Finite Möbius Groups.- Detour in Topology: Euler-Poincaré Characteristic.- Detour in Graph Theory: Euler, Hamilton, and the Four Color Theorem.- Dimension Leap.- Quaternions.- Back to R3!.- Invariants.- The Icosahedron and the Unsolvable Quintic.- The Fourth Dimension.

Reviews

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. (Gunter 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 fur 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)

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. (Gunter 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 fur 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)

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. (Gunter 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 fur 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)

From the reviews of the second edition: <p> Totha (TM)s a ~Glimpsesa (TM) offer selected material that connect algebra and geometry a ] . 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 Kleina (TM)s famous work on the quintic and the icosahedron. (GA1/4nter M. Ziegler, Zentralblatt MATH, Vol. 1027, 2004) <p> The book is intended a and really manages it a to fill undergraduates with enthusiasm to reach the graduate level. a ] 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. a ] 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 fA1/4r Mathematik, Vol. 141 (2), 2004) <p> 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. a ] there are plenty of references for the interested reader who wishes to pursue a particular topic in greater depth. a ] 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) <p> This is the second, much revised and augmented edition of the book originally published in 1998. It intends to close the gap between undergraduate andgraduate studies in number theory, classical geometry and modern algebra. a ] Each of the chapters is a good read and the book adds up to a wholly appealing entity. a ] It can be warmly recommended a ] . I can well imagine that teachers a ] as well as scientists a ] will benefit from this carefully worked-out textbook. (J. Lang, Internationale Mathematische Nachrichten, Vol. 57 (192), 2003)

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