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." ... Read more

Customer Reviews (8)

The "Bible" of Abstract Algebra
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.

Anyhting Left to Say?
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.

It ain't perfect!
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.

A classic of abstract algebra
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.

Classic Book in Algebra
Great Quality. A book that will be in the bibliography of all algebra textbooks. Worth collection. ... Read more

Customer Reviews (1)

A gem of Group Theory! Please reprint it, Springer!
Van der Waerden wrote this book in 1932, about the same epoch of Hermann Weyl's book on group theory, in German. After a staggering gap of 42 (!) years, he prepared this second edition, this time in English. It's probably the cleanest and most rigorous exposition of group theory ever made "for physicists". The quotes were needed because van der Waerden's slant is mainly towards math, as far as mathematical rigour is concerned. But what a clarity! And what a great choice of topics! After a review of basic quantum mechanics in a rather mathematically rigorous and clean style, itstarts with group theory itself. It manages to proof the basic theorems about groups with or without operators, discuss representation theory in the context of observables and symmetries, expose thoroughly Lie groups (translations, rotations and the Lorentz group), spinors and the Dirac equation, and permutations, and introduce briefly molecule spectra, all that in about 200 pages!! No time-wasting like most other books, which seem to "over-wrap" the main core of the subject, making it hard to peel.There is a rather large amount of references to heavy Lie group theory and functionalanalysis' theorems, but here I have to agree with the author that inserting also theirproofs wouldn't add anything profitable, because they would be beyond the book's scope and totally out of context. Even in these exclusions we notice the author's wisdom.

I deeply regret that this masterpiece is out-of-print. I DO insist that Springer should reprint it as soon as possible! ... Read more

Customer Reviews (1)

Bold
Van der Waerden is a delightful historian because he is not afraid of proposing bold hypotheses. His most provocative claim in this book is that all major mathematical cultures have a common origin in the Neolithic period. This common core of mathematics was focused on Pythagoras' theorem and Pythagorean triples.

Although van der Waerden's thesis is surely too simplistic to be literally true, it is nevertheless good fun to view history through his lens, for from it flow neat answers to intractable why-questions, such as:

Why did ancient man care about Pythagoras' theorem? Because it can be used to calculate the duration of a lunar eclipse as a function of the moon's deviation from the ecliptic (p. 32).

Why did ancient man care about proofs of mathematical theorems? If one starts with a problem such as the eclipse problem then one is virtually forced to discover Pythagoras' theorem by proving it, as neither the theorem nor any empirical rule of thumb that could take its place is likely to suggest itself by other means.

Why did ancient man want to generate Pythagorean triples? In order to construct textbook problems on Pythagoras' theorem that have numerically neat solutions (p. 33).

Why did ancient man become interested in ruler-and-compass constructions? For the sake of the constructions of altars (pp. 10-14). "Those who deprive the agni (altar) of its true proportions will suffer the worse for sacrificing" (p. 13), we read in an ancient Indian text which treats a number of construction problems about areas. The Greeks, for their part, of course had the problem of the duplication of the cube, which is also said to have originated as an altar-construction problem (p. 13).

This indicates the point at which the development from the common Neolithic origin split into two branches, driven by textbook texts (Babylonian, Egyptian, Chinese) and ritual geometry (Hindu, Greek) respectively (pp. 66-69).

Of course more mainstream historians will reject van der Waerden's reconstructions, on good grounds, as Knorr does in his excellent review in the British Journal for the History of Science. But I for one will still read books like van der Waerden's, because the same scrupulousness that drives such reviews also drives these respectable historians to shy away from tackling interesting why-questions head on in the refreshing manner of van der Waerden. ... Read more

Product Description
Chapters: Bartel Leendert Van Der Waerden, Otto E. Neugebauer, Leonard Eugene Dickson, Eric Temple Bell, Morris Kline, Dirk Jan Struik, Isaac Todhunter, Ivor Grattan-Guinness, Paul Tannery, Øystein Ore, Stephen Stigler, Colin Mclarty, Anders Hald, Baldassarre Boncompagni, W. W. Rouse Ball, K. V. Sarma, James Franklin, Amir Aczel, T. L. Heath, William Dunham, James R. Newman, Donald A. Gillies, Tom Whiteside, David Fowler, Jack Copeland, Carl Benjamin Boyer, Louis Charles Karpinski, Howard Eves, Florian Cajori, Joseph Dauben, Dietrich Mahnke, Hermann Hankel, Judith Grabiner, Detlef Laugwitz, Georges Ifrah, Karl Menninger, Michael Bernstein, Tsuruichi Hayashi, Asger Aaboe. Source: Wikipedia. Pages: 120. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: Otto Eduard Neugebauer (May 26, 1899 February 19, 1990) was an Austrian-American mathematician and historian of science who became known for his research on the history of astronomy and the other exact sciences in antiquity and into the Middle Ages. By studying clay tablets he discovered that the ancient Babylonians knew much more about mathematics and astronomy than had been previously realized. The National Academy of Sciences has called Neugebauer "the most original and productive scholar of the history of the exact sciences, perhaps of the history of science, of our age." Neugebauer began as a mathematician, turned first to Egyptian and Babylonian mathematics, and then took up the history of mathematical astronomy. In a career of sixty-five years, he largely created our current understanding of mathematical astronomy from Babylon and Egypt, through Greco-Roman antiquity, to India, Islam, and Europe of the Middle Ages and Renaissance. His influence on the study of the history of the exact sciences is profound. Neugebauer was born in Innsbruck...More: http://booksllc.net/?id=737813 ... Read more

Product Description
Chapters: Max August Zorn, Lodovico Ferrari, Alexander Anderson, Hermann Grassmann, Emmy Noether, Bartel Leendert Van Der Waerden, Emanuel Lasker, Franciscus Vieta, Emil Artin, William Kingdon Clifford, Thomas Harriot, Richard Dedekind, Paul Cohn, Rostislav Grigorchuk, James Joseph Sylvester, Leonard Eugene Dickson, Charles Hermite, Jean-Pierre Serre, Benjamin Peirce, Joseph Wedderburn, Rafael Bombelli, Saunders Mac Lane, Garrett Birkhoff, E. H. Moore, Joachim Lambek, Nicolae Popescu, Irving Kaplansky, Colin Mclarty, Melvin Hochster, Yuri I. Manin, Michel Kervaire, Abū Kāmil Shujā Ibn Aslam, Hyman Bass, Carl Wilhelm Borchardt, Abraham Adrian Albert, Otto Hesse, Ernst Steinitz, Suresh Venapally, Ibn Al-Banna Al-Marrakushi, Paul Albert Gordan, Marshall Hall, Raman Parimala, László Rédei, Paolo Ruffini, David Eisenbud, Alexandre-Théophile Vandermonde, Shimshon Amitsur, Philip Hall, Idun Reiten, Eugene Dynkin, Reinhold Baer, Karin Erdmann, Pierre Samuel, Israel Nathan Herstein, Daniel Hershkowitz, Nathan Jacobson, Michael Artin, Cyrus Colton Macduffee, Ruth Moufang, Rosemary A. Bailey, Taro Morishima, Alexander Skopin, Mitrofan Cioban, Peter Cameron, Heinrich Martin Weber, Roger Wolcott Richardson, Dan Segal, Eben Matlis, Guy Terjanian, Uwe Storch, Bjarni Jónsson, Tadashi Nakayama, Viktor Wagner, Kiiti Morita, George Jerrard, Archibald Read Richardson, Paul Monsky, György Hajós, Tibor Szele, Erland Samuel Bring. Source: Wikipedia. Pages: 334. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: Emanuel Lasker (December 24, 1868 January 11, 1941) was a German chess player, mathematician, and philosopher who was World Chess Champion for 27 years. In his prime Lasker was one of the most dominant champions, and he is still generally regarded as one of the strongest players ever. It is often said that L...More: http://booksllc.net/?id=244649 ... Read more

Customer Reviews (2)

Partial synopsis
Early Egyptian astronomy seems to have been of a practical origin. It was noticed that Sirius was the "herald of the flood" (p. 8). "The flooding of the Nile over its banks is the most important event in the Egyptian agricultural year. It gives new life to the parched land. This event is heralded some weeks beforehand by a striking event in the firmament, namely the first visibility of Sirius in the morning sky." (p. 9). Other practical advice based on stars include: "when strong Orion begins to set, then remember to plough"; and "fifty days after the solstice is the right time for men to go sailing" (p. 12). The stars were also used to tell time at night. "In the course of the centuries these Stars of Time became Gods of Time and Destiny." (p. 14). "'From their might derives everything that humanity encouters in the way of disasters,' says the revelation of Hermes Trismegistos." (p. 29). "According to Hermes Trismetgistos the decans can also be called 'horoskopoi'---hour indicators. The decan that rises in the hour of the birth of a child determines the nature of the child." (p. 32).

Babylonian astronomy, on the other hand, seems to be linked to, and largely dictated by, astrology as far back as the record goes. "The oldest cuneiform texts giving the positions of the planets in the zodiac date from the second half of the fifth century B.C. To just this period, and to Babylon too, belongs the oldest horoscope that has been preserved." (p. 2). Of course Babylonian astronomy is much older than this, but precise knowledge of planetary positions were not important as long as astrology was impersonal, perhaps for the reasons given below. Indeed, "Old-Babylonian astrology was not interested, or at least not in the first place, in the fate of the individual. Its principal interest was the well-being of the country. Its predictions concern the weather and the harvest, drought and famine, war or peace and of course also the fate of the Kings." (pp. 48-49).

The rationale for impersonal astrology may have included the following. "Just as the great Gods Sin (the moon) and Shamash (the sun) are obviously responsible for the regular procession of months, days and years, and thus influence our entire life, so it was thought that the Goddess Ishtar [Venus] communicates important things to us by her appearances and disappearances." (p. 57). Above we saw some examples of apparently important influences of the stars, in the spirit of which one will say things like "O Ursa major ... Put truth for me" (p. 58), as one prayer reads. A further consideration is the plausibility of the idea of a strictly periodic universe (of course the world would be periodic if it was determined by the heavens, which are paradigmatically periodic). As Eudemos was later to relate, "If we are to believe the Pythagoreans, I shall in the future, even as everything recurs according to the Number, again tell you tales here, holding this little stick in my hand, while you will sit before me as you do now; and likewise everything else will be the same." (p. 114).(Pythagoras' conception of the world owes much to Babylonian ideas, especially so his emphasis on number, which we shall see below is a very dear concept to the Babylonians. The periodicity at which the world repeats is presumably a common multiple of all planetary periods.)

The rationale for individual astrology seems to have included the following. The idea that the souls of the dead rise to the heavens is an old one. Not the first example is that "the inscription for the fallen at the battle of Potidea (-431) says: 'The aether will receive their souls, as the earth receives their bodies'" (p. 146). From here it is a rather short step to the idea that, as expressed for example "in Servius' commentary on Aeneid VI 714, the souls before birth go down through the planetary spheres, acquiring thereby from Saturn inertia, from Mars wrath, from Venus lust, from Mercury avarice, from Jupiter ambition" (p. 144). Another argument in support of this view is that the heavens are the paradigm of self-motion, which is not displayed by soulless objects. As Plato puts it: "the soul which has lost its wings is borne along until it gets hold of something solid, ... taking upon itself an earthly body, which seems to be selfmoving, because of the power of the soul within it" (p. 147, Phaidros 246b-c).

As for mathematical astronomy, the Babylonian theory was decidedly arithmetical and instrumental as opposed to geometrical and realist. The foundation of the theory is periodicity data, which can be extremely accurate by averaging many years worth of observations. For example, the Babylonian value for the average lunar period is accurate to the second (p. 240). "Linear zigzag functions" fitted to such data formed the foundations of the theory, though ad hoc corrections were introduced whenever expedient, such as changes in slope or extra zigs and zags. An illustrative example of the commitment to this arithmetical-intrumeltal view is the fact that even eclipse magnitude was sometimes modelled by such a function (p. 239), even though the function is obviously nonsensical on most of its domain (viz. when there are no eclipses). Because of its character the theory was therefore remarkably accurate on observable, approximately zigzag-periodic phenomena (such as the positions of the sun and the moon) and phenomena derivable therefrom (such as lunar eclipses, though this was sometimes treated as a primitive observable in itself). The theory was however "not very good" (p. 278) on problems for which a geometric understanding would have been beneficial, such as planetary positions. For the same reason, the theory was "not very good for solar eclipses, because the Babylonians had no means of calculating the lunar parallax, which has a considerable influence on the magnitude of a solar eclipse" (p. 120). This is ironic since there is a famous story about Thales predicting a solar eclipse (which he must almost certainly have done on the basis of Babylonian theory). The ancient sources tell us that Thales predicted only that a solar eclipse would occur in a particular year, but this crude prediction was apparently sufficient to impress his countrymen (pp. 120-122).

price out of this world
This is one of the finest books on the history of early astronomy I have ever read.It is highly recommended.Oxford University Press published it years ago in hardcover (the edition I read).I wanted to reacquire it but when I saw the price my jaw dropped !
It covers early egyptian astromony (what little there is of it)and does a masterful job on babylonia .
The price however, prohibits me recommending it to anyone, as it is possible to cover the subject as a much lower cost. ASTRONOMY BEFORE THE TELESCOPE , FROM THE OMENS OF BABYLON and ANCIENT ASTRONOMY AND CELESTIAL DIVINATION are three other books that come to mind, the latter two dealing exclusively with Babylonia.For Egypt there is Lockyer's DAWN OF ASTRONOMY.The list goes on.
Still, it is a first rate study of the subject. ... Read more

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Customer Reviews (3)

Nice collection of papers leading to quantum revolution, but some might feel discouraged reading
It's no wonder some might feel frustrated or discouraged reading the papers in this collection. Even though those papers were written several decades ago, they had been all forefront research papers then. Some papers should be difficult even for a physics major if one is not in the specific field; some are difficult because of the usage of "old-style" notations such as writing matrix equations in a certain way; still you may find a couple papers very much readable even with a minimal amount of training in mathematical skills.

Over My Head
I have read several dozen books on the subject of Cosmology and related topics. This is a technically oriented book filled with intricate mathematical formulas and is clearly geared for advanced students. I am not shy about mathematics or formulas as a rule and have handled other books on Quantum mechanics, relativity and physics but this book was just over my head.

If you really want to understand Quantum Mechanics...
...you're probably out of luck, because no seems to really understandQuantum Mechanics! However, understanding how these very strange conceptsarose while physics was "under construction" in the early 20thCentury is probably the best way to come to terms with it. This book seemsto be the best thing to a "blow by blow" account of how differentideas emerged, were discussed, and were modified or rejected. It containstranslations of many of the original (mostly German) key papers, along witha prefatory essay that places them in context. Reading these papers is muchpreferable to reading the typical brief history of QM presented in mosttext books: you can see what the pioneers were really thinking about, intheir own words, as opposed to a retrospective point of view that ignoresthe ambiguities they actually faced.

It begins with Einstein'sderivation of the Planck spectral distribution law; includes Ehrenfest'sdiscussion of adiabatic invariants; Bohr's final presentation of the oldQuantum Theory; several papers on the theory of dispersion; and on to thedevelopment of matrix mechanics by Heisenberg, Born and Jordan; and Dirac'sreformulation.

It does not cover Schroedinger's development of wavemechanics, nor the derivation of the Dirac equation for the relativisticelectron, nor quantum field theory. However, the period covered was themost paradigm-shattering part of the development of QM.

Perhapsunfortunately, it is unlikely that the typical student of Physics will havethe time to study this book. However, for those who really love Physics andwant to understand it, this book is essential. With 17 major papers, it hasenough material to occupy months of personal study. ... Read more

Customer Reviews (2)

Very interesting but could be more helpful
This in an opinionated and confident history of Greek mathematics. On issues of interpretation and historiography, a few sections are original and many are based on the classical German literature, not readily available to the modern reader. The problem is with the mathematics, where van der Waerden is very unhelpful. He follows the standard practice of simply reproducing or lightly paraphrasing the original sources. A typical section on Archimedes, which almost entirely lacks commentary, simply reproduces a number of proof and ends "All this is found in Archimedes, in essentially the same words" (p. 220). This is especially ironic since van der Waerden himself portrays as a main reason for the demise of Greek geometry "the difficulty of the written tradition" (p. 266): "An oral explanation makes it possible to indicate the line segments with the fingers; one can emphasize essentials and point out how the proof was found. All of this disappears in the written formulation of the strictly classical style. The proofs are logically sound, but they are not suggestive. One feels caught in a logical mousetrap, but one fails to see the guiding line of thought. As long as there were no interruption, as long as each generation could hand over its method to the next, everything went well and science flourished. But as soon as some external cause brought about an interruption in the oral tradition, and only books remained, it became extremely difficult to assimilate the work of the great precursors and next to impossible to pass beyond it." (p. 266). This is exactly how we feel. It seems obvious, then, that a good history of Greek mathematics will aspire to recreate the qualities here ascribed to the oral traditions. Unfortunately, van der Waerden is generally happy to reproduce the "logical mousetraps" and leave it at that. One further point is important to remember in this context. Today's mathematics tends to be congenial to "logical mousetraps" and shun intuition and applications. In Antiquity it was probably the other way around: it was precisely the mathematicians who were striving for more intuition and applications, while philosophical and other prejudices were exactly opposed to such a development. This inversion of the current roles is indicated in Plutarch: "Plato himself censured those ... who wanted to reduce the duplication of the cube to mechanical constructions, because [it is based on] a non-theoretical method; for in this manner the good in geometry is destroyed and brought to naught, because geometry reverts to observation instead of raising itself above this and adhering to the eternal, immaterial images in which the immanent God is the eternal God." (p. 163). In other words: the surviving mathematical texts are "censured" by non-mathematicans on non-mathematical grounds, whereas the subject matter screams out for a less formalistic treatment.

Make no mistake, this is a math book -
A comparable book might be an Introduction to Algebra book.Complete with descriptions and examples of usage (and of course history).Ifyou're interested in knowing more about Babylonian sexagesimal notation, Egyptian fractions, poetic numbers, Pythagoras, Plato or Hellenistic science interest you then this is a good source for the material and thankfully it has been translated into English.While I believe the translation was accurate, this can be a sleeper if you're not fond of numbers or their manipulation. ... Read more