zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Algebra. 3rd revised ed. (English) Zbl 0984.00001
Graduate Texts in Mathematics. 211. New York, NY: Springer. xv, 914 p. EUR 74.95/net; sFr. 124.50; £52.50; $ 69.95 (2002).

The first edition of this book (1965; Zbl 0193.34701) consisted of three parts: basic concepts, field theory and linear algebra, and as a modern down-to-earth approach with a personal touch it attained great popularity. The second edition (1984; Zbl 0712.00001) added some topics, mainly on commutative algebra and homological algebra.

The current third edition has grown again, the additions dealing with topics close to the author’s heart from number theory, function theory and algebraic geometry. For the math graduate who wants to broaden his education this is an excellent account; apart from standard topics it picks out many items from other fields: Bernoulli numbers, Fermat’s last theorem for polynomials, the Gelfond-Schneider theorem and (as an exercise, with a hint) the Iss’sa-Hironaka theorem. This makes it a fascinating book to read, but despite its length it leaves large parts of algebra untouched. Semisimple algebras get a very cursory treatment (no mention of crossed products or the Brauer group) and there is only the merest trace of Morita theory; there are no Ore domains, Goldie theory or PI-theory. Graphs, linear programming and codes, constructions like ultraproducts and Boolean algebras are also absent, and lattices are only of the number-theoretic sort (reseaux, not treillis).

Bearing these limitations in mind, the reader will nevertheless find a very readable treatment of many modern mainline topics as well as some interesting out-of-the-way items.

Editorial comment: Note that there is also a 3rd ed. published by Addison-Wesley 1993 reviewed in Zbl 0848.13001.


MSC:
00A05General mathematics
12-01Textbooks (field theory)
13-01Textbooks (commutative algebra)
15-01Textbooks (linear algebra)
16-01Textbooks (associative rings and algebras)
18-01Textbooks (category theory)
20-01Textbooks (group theory)
14-01Textbooks (algebraic geometry)
11-01Textbooks (number theory)