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Algebra, Volume I

B. L. van der Waerden
Springer Verlag
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Fernando Q. Gouvêa
, on

In the glad to have you back department, I’m delighted that Springer has decided to reprint the two volumes of B. L. van der Waerden’s Algebra. Based on lectures by Emmy Noether and Emil Artin, this is the book that brought “abstract algebra” to the mathematical world. It is not for the faint of heart: van der Waerden is terse and precise: as he says, “knapp (aber lückenlos).” He does not spend much time on easy details.

On the other hand, the book reflects the excitement that accompanied the birth of axiomatic algebra. It was written when the power of the new methods was still news for most mathematicians, and a guide to the new ideas was still needed. This is that guide. It covers all the usual topics and then some. Not, perhaps, a book to give to an undergraduate (though who knows? for some undergraduates, it might just work), but still a book to treasure. I’m glad it’s back. My only hesitation is to wonder whether it would not have been better to bring it back as one fat volume instead of two thin ones.

Fernando Q. Gouvêa is the editor of MAA Reviews.

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. The Galois Theory; 9. Ordering and Well Ordering of Sets; 10. Infinite Field Extensions; 11. Real Fields; Index.