Geometry of the Quintic [Paperback]

Jerry Shurman (Author)

Book Description

ISBN-10: 0471130176 | ISBN-13: 978-0471130178 | Publication Date: January 31, 1997 | Edition: 1

A chance for students to apply a wide range of mathematics to an engaging problem

This book helps students at the advanced undergraduate and beginning graduate levels to develop connections between the algebra, geometry, and analysis that they know, and to better appreciate the totality of what they have learned.

The text demonstrates the use of general concepts by applying theorems from various areas in the context of one problem--solving the quintic. The problem is approached from two directions: the first is Felix Klein's nineteenth-century approach, using the icosahedron. The second approach presents recent works of Peter Doyle and Curt McMullen, which update Klein's use of transcendental functions to a solution through pure iteration.

Filling a pedagogical gap in the literature and providing a solid platform from which to address more advanced material, this meticulously written book:
* Develops the Riemann sphere and its field of functions, classifies the finite groups of its automorphisms, computes for each such group a generator of the group-invariant functions, and discusses algebraic aspects of inverting this generator
* Gives, in the case of the icosahedral group, an elegant presentation of the relevant icosahedral geometry and its relation to the Brioschi quintic
* Reduces the general quintic to Brioschi form by radicals
* Proves Kronecker's theorem that an "auxiliary" square root is necessary for any such reduction
* Expounds Doyle and McMullen's development of an iterative solution to the quintic
* Provides a wealth of exercises and illustrations to clarify the geometry of the quintic

Editorial Reviews

From the Publisher

A consolidation of mature mathematical subjects like geometry, linear algebra, group theory, complex analysis and Galois theory into one source, this simple, easy-to-follow text develops deep connections between seemingly unrelated areas in mathematics. It updates Felix Klein's "Lectures on the Icosahedron and Equations of the Fifth Degree", and Peter Doyle's and Curt McMullen's "Solving the quintic by iteration." It provides an active approach to learning, and presents familiar subjects in a nonredundant, forward looking fashion.

From the Back Cover

A chance for students to apply a wide range of mathematics to an engaging problem

This book helps students at the advanced undergraduate and beginning graduate levels to develop connections between the algebra, geometry, and analysis that they know, and to better appreciate the totality of what they have learned.

The text demonstrates the use of general concepts by applying theorems from various areas in the context of one problem—solving the quintic. The problem is approached from two directions: the first is Felix Klein's nineteenth-century approach, using the icosahedron. The second approach presents recent works of Peter Doyle and Curt McMullen, which update Klein's use of transcendental functions to a solution through pure iteration.

Filling a pedagogical gap in the literature and providing a solid platform from which to address more advanced material, this meticulously written book:

• Develops the Riemann sphere and its field of functions, classifies the finite groups of its automorphisms, computes for each such group a generator of the group-invariant functions, and discusses algebraic aspects of inverting this generator
• Gives, in the case of the icosahedral group, an elegant presentation of the relevant icosahedral geometry and its relation to the Brioschi quintic
• Reduces the general quintic to Brioschi form by radicals
• Proves Kronecker's theorem that an "auxiliary" square root is necessary for any such reduction
• Expounds Doyle and McMullen's development of an iterative solution to the quintic
• Provides a wealth of exercises and illustrations to clarify the geometry of the quintic

See all Editorial Reviews

Product Details

• Paperback: 216 pages
• Publisher: Wiley-Interscience; 1 edition (January 31, 1997)
• Language: English
• ISBN-10: 0471130176
• ISBN-13: 978-0471130178
• Product Dimensions: 10 x 7 x 0.6 inches
• Shipping Weight: 13.8 ounces
• Average Customer Review: 4.0 out of 5 stars  See all reviews (2 customer reviews)
• Amazon Best Sellers Rank: #3,053,505 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

13 of 13 people found the following review helpful:

5.0 out of 5 stars Felix Klein and polynomials of the fifth degree, December 30, 2000

By A Customer

This review is from: Geometry of the Quintic (Paperback)

There are many books that discuss the fact that you cannot solve the general polynomial of the fifth degree (i.e. a quintic) using radicals. They often mention that Felix Klein showed there is a method to find the roots using rotations of the icosahedron, the regular polyhedron with 20 triangular faces. It is hard to find out exactly what this means (Klein's own book is out of print). Shurman gives all the details, and is a well written book combining group theory and geometry.

In brief, Klein's result goes like this: Find all rotations that leave the icosahedron invariant, which turns out to be isomorphic to A5, the alternating group on 5 letters. Use stereographic projection to map the sphere onto the plane, and use this to map the rotations fixing the icosahedron to a group G of linear fractional transformations. Next find an icosahedral invariant f, which is a rational function f(z) (which turns out to have degree 60) invariant under G. That is f[(az+b)/(cz+d)] = f(z) for all transformation z -> (az+b)/(cz+d) in G. Finally, let g(w) = z be the inverse function to f(z) = w. Then Kleins' result is that for any quintic, there is a formula that gives its roots as an expression involving the coefficients of p, radicals, and the function g().

If this doesn't make a lot of sense, it will after reading Shurman's book. He starts at the beginning in chapter 1 by explaining how to map the sphere onto the plane using stereographic projection. Chapter 2 computes the five regular polyhedra and their rotation groups, giving explicit generators for each group. Chapter 3 computes invariant functions, rational functions preserved by groups of linear fractional transformations. Chapters 4 and 5 complete the explanation of how to solve the quintic via the icosahedron, and chapters 6 and 7 treat some related topics.

The book has lots of explicit computations. As just one example: after Shurman proves that the rotations of the icosahedron can be represented as a unitary group, he computes the actual matrices that generate the group.

Many key parts of proofs are left as exercises, but they are almost all easy. The book is unusually well proof-read. I only noticed one misprint: one page 95 line 5, there is a - that should be a +.

I greatly enjoyed this book. I found it be a very pleasant read combining basic abstract algebra and Euclidean geometry.

3 of 4 people found the following review helpful:

3.0 out of 5 stars Caveat emptor!, December 13, 2001

By A Customer

This review is from: Geometry of the Quintic (Paperback)

Look at this book before you buy it. The author gets five stars; the publisher, one at most. What a shame! This is just the book for the summer before you start grad school. See Galois theory in action! But check it out of the library. It's printed on blotting paper. The illustrations are done in shades of black. If you wear glasses you will think they are dirty, but sadly, no amount of cleaning will make this book look as sharp and clear as its ideas. Of course, what you really study will be in your own handwriting, but a book this expensive should be beautiful.