Customer Reviews

Algebra: Volume I

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mathematically, this book is first rate. I turn to it when stuck on a passage in hungerford. However, this book shows its age in certain respects - e.g. category theory is completely snubbed - hence the index doesn't square well with a lot of the current standard terminology.

Still, the fact that this book is in its seventh english printing says something about its value. It's like reading Dirac's principles of QM: sure, Griffiths is much easier, and exactly covers the standard undergrad subject matter -- but Dirac does it all in a third of the space, and with such brilliant clarity...

Bottom line: don't buy this as your first algebra book, because it's old-fashioned. The point of reading a textbook is to get you far enough out there that you can read the current literature and be done with textbooks. But, if you already own Dummit and Foote (for undergrad material, lots of hand-holding in tough parts) and Lang or Hungerford (or even Herstein) for the standard first-year grad course perspective, this will round out your algebra library nicely.

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 theory'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. Automorphism 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.

An excellent book for putting it all together, but not for self-study -- it is too "thin" for that, giving in many cases just outlines of proofs. For the mathematically mature reader only. Two volumes.

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.

I think there are few words to say about this book. This is a classic of Abstract Algebra very well known around the world among algebrists. This is a book that everybody interested about Algebra must read.

Great Quality. A book that will be in the bibliography of all algebra textbooks. Worth collection.

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 had these problems.

if you want to learn algebra, you need this book at first, necessary thing, and you will feel that is true even after you read the first pages only, that is very nice book, also this can give you imagine about modern algebra, the book constructs whole algebra by strict way, finally, one word about this, if you reference this book in your scientific work or student work, this can be proof of that your job is more strong, and in more higher level, ok,