Customer Reviews

6
4.3 out of 5 stars
5 star
4
4 star
1
3 star
0
2 star
1
1 star
0
Your rating(Clear)Rate this item


There was a problem filtering reviews right now. Please try again later.

20 of 21 people found the following review helpful
on July 10, 2008
Format: PaperbackVerified Purchase
This is a wonderfully written, highly accessible introduction to polynomial equations (quadratics, cubics, etc.) and their solutions by radicals, leading step by step to an introduction to Galois theory.

Galois theory is presented only towards the end of the book. Readers already familiar with the solutions of quadratic equations, depressed cubics, cubics, and quartics will find the first half of the book somewhat redundant. But it is nevertheless very pleasant to read, with succinct notes on the historical background, and (mostly) self-contained short sections.

It reads very well all the way to the end. It gets a little harder when Galois theory is introduced. But that's perhaps to be expected. I can't say that I master the subject, but certain things (about polynomial equations) are a great deal clearer for me now.

I do have one reservation (but I did not knock off a star for that): the editing (of this English translation of the German original) is quite poor: there is a typo just about every other page. I am very sensitive to typos, and most readers probably won't (nor should they) care -- but there are some typos in the math here and there, and that's plain unacceptable.
0CommentWas this review helpful to you?YesNoSending feedback...
Thank you for your feedback.
Sorry, we failed to record your vote. Please try again
Report abuse
9 of 10 people found the following review helpful
on January 4, 2010
Format: Paperback
This is a very interesting and entertaining book. It allows the student of Galois Theory to 'look under the hood': the modern day presentation of that theory is essentially Emil Artin's streamlined field theory approach, which is a beautiful theory, but many students would appreciate more detail about how mathematicians went from solving polynomial equations to analyzing field extensions.

The book goes a good way toward filling this gap. By providing appropriately chosen concrete examples, the author leads the reader to a deeper understanding of the nuts and bolts underlying Galois Theory (and to some pretty lengthy -- but worthwhile -- computations -- by working the exercises at the end of each chapter). The book also reveals how mathematical ideas evolve and how close Lagrange and Ruffini came to the (still revolutionary) ideas of Galois. The author keeps prerequisites at a minimum, but he does make demands upon the 'beginner'. The more advanced asides are appropriately placed throughout the book and can be skipped without consequence (the reader will want to return to them on a second reading, though). The book begins with the historical methods used both to solve cubic and quartic polynomial equations as well as to reduce and solve special polynomial equations of higher degree. The book culminates in Galois's original 'elementary' view of what is now called the Galois group of the solutions of a polynomial equation, followed by the correspondence between the 'decomposition' of a such a group into its subgroups and the present day field extensions (after a minimal introduction to groups and fields). In addition to the historical detail, there are many asides of further explanation or further computational techniques as well as references to the literature.

I have three 'stylistic' qualms about the book: a lack of evenness in the presentation (the German edition is better at demarking the asides with gray box lines); the lack of some steps (and therefore clarity) for the beginner in some of the more advanced asides; and the many typos, but the author does maintain an up-to-date list of errata on his website ([...]).
0CommentWas this review helpful to you?YesNoSending feedback...
Thank you for your feedback.
Sorry, we failed to record your vote. Please try again
Report abuse
17 of 21 people found the following review helpful
on January 2, 2009
Format: PaperbackVerified Purchase
When I see "for Beginners" in the title of a math book, my expectation is that the proofs of theorems will contain many simple steps and build on well-formulated definitions. This book did not meet those expectations. The proofs were terse and hard to follow, and the nomenclature was stilted (probably a result from translating the original German prose). The first few chapters were fine, and at times even interesting, but from chapter 4 onward, the text became unreadable for a true beginner. Very disappointing, especially considering the book's steep price.
33 commentsWas this review helpful to you?YesNoSending feedback...
Thank you for your feedback.
Sorry, we failed to record your vote. Please try again
Report abuse
4 of 4 people found the following review helpful
Format: PaperbackVerified Purchase
There are two ways to approach the teaching of a certain area of mathematics: the formal and one that emphasizes intuitive understanding with historical motivation. Formal works of mathematics are the majority, and in all of these one can see the full power of mathematical rigor and abstraction. But these are lacking in getting the reader to appreciate the subject, and it is very difficult to accept how the essential ideas were actually thought of. In the minority are those works that attempt to grant insight to the reader who craves for a more in-depth view of the mathematical concepts. These books are probably so rare because of the emphasis on rigor in mathematics and because they are much more difficult to write than formal texts and books. And it is insight that makes a great mathematician.

This book on Galois theory is of the latter class, because of its emphasis on historical motivation and the many concrete examples given between its covers. The author has done a fine job of relating to the reader just how Galois theory arose and why its form as Galois discovered it, is very different than what one will find in modern books on the subject. Galois definitely was a "modern" mathematician in the sense that he emphasized studying mathematical objects according to the transformations they can support. This paradigm dominates contemporary pure mathematics, leaving applied mathematicians the worry of how to extract reality and numbers from highly esoteric constructions and theories.

As the author explains brilliantly and originally, it was the desire to find solutions of higher degree polynomials in terms of radicals that motivated Abel and Galois to investigate to what extent this is possible. But before reading this book most readers will already have known this reason for Galois theory. What the author brings to the book is an appreciation of the efforts and failures in finding a general formula for the solution of the quintic equation, either by analogs of completing the square or by using certain transformations that simplified the equation. Readers will also learn of the resistance to negative numbers and complex numbers when they were first proposed, and how ruler and compass constructions can be expressed in terms of purely algebraic manipulations.

It is amazing to read also that Galois' ideas were not accepted right away, taking a couple of decades before they were appreciated and only because a certain mathematician advocated them and eventually got them into print. By modern standards Galois would be labeled as an amateur, but given the impact of his ideas, he finds himself immortalized, and modern algebra would probably not have the form it does without him.
0CommentWas this review helpful to you?YesNoSending feedback...
Thank you for your feedback.
Sorry, we failed to record your vote. Please try again
Report abuse
1 of 3 people found the following review helpful
on January 23, 2014
Format: PaperbackVerified Purchase
This is a very readable book for the non-mathematian (but you need a freshman level algebra knowledge). He has a sense of humor, which makes it pleasant to read, and the historical perspective dilutes out the deadly serious stuff so you are motivated to move ahead. It is still a very serious book, if you read it carefully you will know a great deal about algebra in general and Galois groups in particular.
0CommentWas this review helpful to you?YesNoSending feedback...
Thank you for your feedback.
Sorry, we failed to record your vote. Please try again
Report abuse
19 of 38 people found the following review helpful
on November 4, 2007
Format: PaperbackVerified Purchase
The main focus of Jorg Bewersdorff's "Galois Theory for Beginners: An Historical Perspective" is "...[polynomials] and their solutions..." Chapters 1 through 3 are about classical methods (formulas) for solutions on cubic and bi-quadratic polynomials. Chapter 4 and 5 are "systematic investigation of the...solution formulas..." Chapter 6's primary topic is "...[polynomials] that can be broken down into...lower degree." Chapter 7 focuses on the construction of regular polygons with straightedge and compass. Chapter 8 is about "...finding a general solution formula for [the] fifth-degree [polynomials]..." Chapter 9 and 10 focus on Galois theory: "...the limits of solvability of [polynomials] in radicals."

I began my pursuit "...to understand why a general...[polynomial] of the 5th degree should have no solution in radical..." decades ago, as if Jorg Bewersdorff. His book is the best I have ever read on Galois theories. For instances:

1. Theorem 10.18 An [polynomial] is solvable in radicals, that is all of its solutions can be expressed in terms of nested roots whose radicands can be expressed in terms of the coefficients using the four basic operations, if and only if its Galois group is solvable...

2. Definition 9.2 For a [polynomial] without multiple solutions whose coefficients lie in a field K, the Galois group (over the field K) is the set of all permutations s in the symmetric group Sn that permute the indices 1,...,n of the solutions x1,...,xn in such a way that for every polynomial h(X1,...,Xn) with coefficients in K and h(x1,x2,...,xn) = 0, one has h(x(s(1)),...,x(s(n))) = 0 ...

3. Definition 10.17 A finite group G is called solvable if there is a chains of groups {id}=G0 ( G1 ( G2 ( ... ( G(k-1) ( G(k) = G for which the subgroup G(j) is a normal subgroup of the next group in the chain G(j+1), such that the quotient group G(j+1)/G(j) is cyclic of prime order n...

4. A point in a plane is constructible if and only if its "...coordinates can be expressed in rational numbers and nested square roots using the four basic arithmetic operations (+, - , * , /).
0CommentWas this review helpful to you?YesNoSending feedback...
Thank you for your feedback.
Sorry, we failed to record your vote. Please try again
Report abuse
     
 
Customers who viewed this also viewed


A Book of Abstract Algebra: Second Edition (Dover Books on Mathematics)
A Book of Abstract Algebra: Second Edition (Dover Books on Mathematics) by Charles C. Pinter (Paperback - January 14, 2010)
$11.52
 
     

Send us feedback

How can we make Amazon Customer Reviews better for you?
Let us know here.

Your Recently Viewed Items and Featured Recommendations 
 

After viewing product detail pages, look here to find an easy way to navigate back to pages you are interested in.