| Date: 28 April 2011, 07:14
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Algebra: Volume I
By B.L. van der Waerden
* Publisher: Springer
* Number Of Pages: 265
* Publication Date: 2003-10-21
* ISBN-10 / ASIN: 0387406247
* ISBN-13 / EAN: 9780387406244
Product Description:
This widely known textbook, formally titled Modern Algebra, by the noted Dutch mathematician van der Waerden is now back in print. Algebra originated from notes taken by the author from Emil Artin's lectures. The author extended the scope of these notes to include research of Emmy Noether and her students. The first German edition appeared in 1930-1931, with subsequent editions having been brought up to date. "The basic notions of algebra, groups, rings, modules, fields, and the main theories pertaining to these notions are treated in the classical two volume textbook of van der Waerden. Although more than half a century has elapsed since the appearance of this remarkable book, it is in no way dated, and for the majority of the questions it treats, no better source can be found even today." (I.R. Shafarevich: Encyclopaedia of Mathematical Sciences, Volume 11.1990)
Summary: The "Bible" of Abstract Algebra
Rating: 5
There are millions of Christian books to explain God's Words, but the best book is still The Bible.
Isomorphically, this book is the "Bible" for Abstract Algebra, being the first textbook in the world (@1930) on axiomatic algebra, originated from the theroy's "inventors" E. Artin and E. Noether's lectures, and compiled by their grand-master student Van der Waerden.
It was quite a long journey for me to find this book. I first ordered from Amazon.com's used book "Moderne Algebra", but realised it was in German upon receipt. Then I asked a friend from Beijing to search and he took 3 months to get the English Translation for me (Volume 1 and 2, 7th Edition @1966).
Agree this is not the first entry-level book for students with no prior knowledge. Although the book is very thin (I like holding a book curled in my palm while reading), most of the original definitions and confusions not explained in many other algebra textbooks are clarified here by the grand master.
For examples:
1. Why Normal Subgroup (he called Normal divisor) is also named Invariant Subgroup or Self-conjugate subgroup.
2. Ideal: Principal, Maximal, Prime.
and who still says Abstract Algebra is 'abstract' after reading his analogies below on Automorphism and Symmetric Group:
3. Automatism of a set is an expression of its SYMMETRY, using geometry figures undergoing transformation (rotation, reflextion), a mapping upon itself, with certain properties (distance, angles) preserved.
4. Why called Sn the 'Symmetric' Group ? because the functions of x1, x2,...,xn, which remain invariant under all permutations of the group, are the 'Symmetric Functions'.
etc...
The 'jewel' insights were found in a single sentence or notes. But they gave me an 'AH-HA' pleasure because they clarified all my past 30 years of confusion. The joy of discovering these 'truths' is very overwhelming, for someone who had been confused by other "derivative" books.
As Abel advised: "Read directly from the Masters". This is THE BOOK!
Suggestion to the Publisher Springer: To gather a team of experts to re-write the new 2010 8th edition, expand on the contents with more exercises (and solutions, please), update all the Math terminologies with modern ones (eg. Normal divisor, Euclidean ring, etc) and modern symbols.
Summary: Anyhting Left to Say?
Rating: 5
This book covers a whole lot of subjects in not-so-many pages. As someone pointed before, it is not intended as a first book on the subject. For one thing: there is not many examples on each topic, the exercises require you to really think and solve a problem, rather than introduce further easy examples to fix the concepts taught. My own experience is, I was puzzled first by the level of abstraction, and the lack of concrete examples on 'foreign' topics (at that time) was a little frustrating. Kind of "So what's the big deal with an ideal being principal or not? What's this all about?". After reading other, slower paced books on some of the same topics, van der Waerden becomes clear. I stringly recommend Elements of Number Theory and Elements of Algebra by John Stillwell, and Serge Lang's Linear Algebra (Undergraduate Texts in Mathematics) before attacking this one.
That said, I do not regret buying this book at all. On the contrary, the first frustration became a strong motivation to complement it; and on the way I discovered a whole wonderful world.
Summary: It ain't perfect!
Rating: 3
OK, it's a classic. Still, I've got complaints.
Consider this:
A Euclidean ring is defined in van der Waerden's "Algebra" in such a way that the reals are a Euclidean ring. Just define g (the norm) as a constant. Since every number has an inverse, the division algorithm is satisfied since we can always have a remainder of zero. Fine. No problem.
Now, half a page under the definition of Euclidean ring, we have a discussion about "the" greatest common divisor of two elements, a, and b, of a Euclidean ring. The 'definition' of the term 'greatest common divisor' is given:
" ... d is also the 'greatest common divisor'; that is, all common divisors of a and b are divisors of d."
OK. Fine. Now, consider the reals which are a Euclidean ring by the definition given here (and I've seen similar elsewhere). Every non-zero real is a common divisor of every pair of reals. Furthermore, every non-zero real divides every one of these common divisors, so every common divisor is a greatest common divisor. That is, every non-zero real is a greatest common divisor of every pair of real numbers.
Well, this is not inconsistent, but the term 'greatest common divisor' in this case, is not descriptive to say the least. Furthermore, the description of a number fitting the definition of greatest common divisor as 'the' greatest common divisor is worse. It is, in this case, wrong.
So we have a mess. The difficulty would go away if we could not make fields fit the definition of Euclidean ring.
Here's another one:
"An ideal in D is called 'maximal' if it is not included in any other ideal in D except D itself, ...". OK, at this point, it sounds like D is a maximal ideal, but maybe not, depending on exactly what is meant by "... other ... except..." (although, that D's exclusion is implied by these words is far from clear and one wonders why, if it is intended that D be excluded, it is not made explicit).
However, the definition continues with an alternate wording, "... or in other words, if it has no proper divisors except the unit ideal D." OK, so this recasting excludes D itself if it is taken to mean that it is required that D be an exceptional proper divisor, but again, this is far from clear. But then the implication that the term 'maximal ideal' includes the ring D itself is strengthened in the statement of the theorem which follows immediately: "Any maximal ideal p in D, different from D itself, ...".
Well if D is not supposed to be maximal, why put in the unnecessary words "different from D itself"?
We are given a very ambiguous idea of 'maximal ideal' here. In definitions given by others, 'maximal ideal' unambiguously excludes the ring D, itself, which is better.
These are not the only problems of this sort.
Still, the book is very interesting. As an early translation, these kind of problems are forgiveable. I would hope a modern text on the settled, well understood material covered in van der Waerden's text would not have such problems. Unfortunately, I find that most texts covering well understood, settled material do have such problems, and it is a rare gem that does not.
It takes a lot more time to read a book with difficulties like those described above, time that could be devoted to learning something else.
I wonder whether the original German text
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