Algebra. Volume I. Based in part on lectures by E. Artin and E. Noether. Transl. from the German 7th ed. by Fred Blum and John R. Schulenberger. 1st paperback ed. (English) Zbl 1032.00001

New York, NY: Springer. xiv, 265 p. (2003).
Almost three quarters of a century ago, precisely in 1930, B. L. van der Waerden published the first edition of his epoch-making, two-volume textbook “Modern Algebra” (1930; JFM 56.0138.01). Based upon lectures given by Emmy Noether and Emil Artin at the University of Göttingen, Germany, in the 1920s, this text was intended as a systematic introduction to the novel abstract, axiomatic approach to algebraic structures, which had just been elaborated in the past three decades before. Combining E. Noether’s pioneering conceptual and structural insights, E. Artin’s unique clarity and elegance of exposition, and his own tremendous gift for synthesis, B. L. van der Waerden set up a textbook that literally revolutionized the teaching of graduate algebra at universities worldwide. More than that, both his unified structural approach and his eloquent, clear, terse and elegant style set the standard for mathematical texts in the sequel.
For many decades after its first appearance, van der Waerden’s text has been the basic source for generations of algebraists, researchers in related fields, and students at German-language universities. Moreover, van der Waerden’s approach to modern algebra has strongly influenced all the textbooks in the field that have been published until now, in one way or another, and still today this outstanding text is mentioned as a first-rate reference and source.
Over the years, van der Waerden’s algebra text has seen seven editions and numerous reprints of them. Apart from many alterations, rearrangements, up-datings and enlargements between 1930 and 1966, B. L. van der Waerden has changed the title of his celebrated two-volume text to simply “Algebra” (1955; Zbl 0067.00501, Zbl 0067.00502), and its overdue English translation finally appeared in 1970, in fact from the last revised German edition (1966; Zbl 0137.25403).
The book under review is a reprinted version of the original English translation of the first volume of B. L. van der Waerden’s “Algebra”, without any alterations. However, it is the first softcover printing, worth the price and particularly handy. It is very gratifying to have such an edition available, too, especially in the case of van der Waerden’s timeless classic, so that further generations of students can both afford it and use it as still one of the best sources for their march into abstract algebra.
Now as before, this first volume contains the first eleven chapters of the whole text, covering the following fundamental topics:
1. Numbers and sets; 2. Groups; 3. Rings and fields; 4. Vector spaces and tensor spaces; 5. Polynomials; 6. Theory of fields; 7. Continuation of group theory; 8. Galois theory; 9. Ordering and well-ordering of sets; 10. Infinite field extensions; 11. Real fields.
As one can see, this still represents a broad range of material of basic abstract algebra, and most contemporary algebra textbooks display about the same arrangement! No doubt, van der Waerden’s “Algebra” is one of the most influential textbooks in mathematics of the 20th century.
For Vol. 2 see the following review (Zbl 1032.00002).


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