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Algebra. Volume II. Based in part on lectures by E. Artin and E. Noether. Transl. from the German 5th ed. by John R. Schulenberger. 1st paperback ed. (English) Zbl 1032.00002
New York, NY: Springer. xii, 284 p. (2003).
For Vol. 1 see the preceding review (Zbl 1032.00001).
This is the second volume of the English translation of B. L. van der Waerden’s classic textbook “Algebra”. In fact, it represents the first softcover printing of the original translation which, on its part, had first appeared in 1970. Accordingly, this unaltered reprinting contains the remaining nine chapters of the entire two-volume text, that is Chapters 12 to 20, covering the following more advanced and special topics:
12. Linear algebra (of rings and modules); 13. Theory of algebras; 14. Representation theory of groups and algebras; 15. General ideal theory of commutative rings (and Noetherian rings); 16. Ideal theory of polynomial rings; 17. Integral algebraic elements; 18. Fields with valuations; 19. Algebraic functions of one variable; 20. Topological algebra.
Actually, the last six chapters also provide a good deal of affine algebraic geometry and algebraic curves, from the classic point of view, and also this masterly exposition is still worth reading today, especially for beginners in commutative algebra and algebraic geometry. Also from the historical point of view, this second volume is highly interesting and enlightening, as it vividly reflects the first refoundation of algebraic geometry in the 1930s. Although B. L. van der Waerden has revised his text several times after World War II, the more recent developments of commutative algebra, homological algebra, and categorical algebra are, of course, not reflected in his book, but this diminishes its value not in the least. It is van der Waerden’s inimitable style of presenting the principles of modern algebra that has survived all new fashions in algebra, and that is setting the standards of writing even in our days.
Compared to many contemporary, comprehensive textbooks in abstract algebra, it is always amazing to realize what wide range of material B. L. van der Waerden managed to cover, and with what seemingly effortless ease, great elegance and ultimate lucidity he was able to do so (on just 500 pages, altogether).
No doubt, also this second volume of B. L. van der Waerden’s classic “Algebra” has maintained its enduring character, over more than seven decades, although the first volume seems to be the more popular and significant part of the whole text, at least so for undergraduate studies.
And now we have “the van der Waerden” also in paperback form, like many other (perhaps less important) classics, which must be seen as a very late honour for such a long-standing bestseller.

MSC:
00A05 Mathematics in general
13-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra
12-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory
20-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory
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