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1
Zbl 1145.17002
Bourbaki, Nicolas
Elements of mathematics. Lie groups and Lie algebras. Chapters 7--9. Transl. from the French by Andrew Pressley. Paperback reprint of the hardback edition 2005. (English)
[B] Berlin: Springer. xi, 434~p. EUR~53.45; SFR~83.00 (2008). ISBN 978-3-540-68851-8/pbk

See the review of the hardback edition (2005) in Zbl 1139.17002.

MSC 2000:: *17-02 Research monographs (nonassoc. rings and algebras)
22-02 Research monographs (topological groups)
01A75 Collected or selected works

Citations: Zbl 1139.17002; Zbl 0505.22006; Zbl 0329.17002; Zbl 0983.17001; Zbl 0904.17001

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2
Zbl 1145.17001
Bourbaki, Nicolas
Elements of mathematics. Lie groups and Lie algebras. Chapters 4--6. Transl. from the French by Andrew Pressley. Paperback reprint of the hardback edition 2002. (English)
[B] Berlin: Springer. xi, 300~p. EUR~53.45; SFR~83.00 (2008). ISBN 978-3-540-69171-6/pbk

See the review of the hardback edition (2002) in Zbl 0983.17001.

MSC 2000:: *17-02 Research monographs (nonassoc. rings and algebras)
22-02 Research monographs (topological groups)
22E10 General properties and structure of complex Lie groups
22E15 General properties and structure of real Lie groups
22E60 Lie algebras of Lie groups
17B20 Simple and semisimple Lie algebras
20E42 Groups with a BN-pair; buildings
20F55 Coxeter groups
51F15 Reflection groups and geometries

Citations: Zbl 0186.33001; Zbl 0249.22001; Zbl 0483.22001; Zbl 0904.17001; Zbl 0983.17001

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3
Zbl 1139.12001
Bourbaki, Nicolas
Elements of mathematics. Algebra. Chapters 4--7. (Éléments de mathématique. Algèbre. Chapitres 4 à 7.) Reprint of the 1981 original. (French)
[B] Berlin: Springer. vii, 422~p. EUR~47.35/net (2007). ISBN 3-540-34398-9/pbk

Book II of N. Bourbaki's fundamental and widely encyclopedic work ``Éléments de Mathématique'' is titled ``Algebra'' and encompasses (until now) ten chapters, which were successively published between 1943 and 1980, originally in French. The volume under review is the faithful and unabridged reprinting of Chapters 4--7 of Bourbaki's treatise on algebra, whose French original, in this arrangement, was first published in (1981; Zbl 0498.12001). Together with its masterly English translation by {\it P. M. Cohn} and {\it J. Howie} (Zbl 0719.12001), this volume has been reviewed repeatedly in the past, and therefore we may refer to these reviews in order to avoid any superfluous reiteration. However, it should be recalled that Chapters 4--7 were successively written in the 1950s, that is, by the founders of the Bourbaki group themselves, thereby reflecting their former revolutionary spirit and style in its original fashion, and that Bourbaki's ``Algebra'' has been among those parts of their monumental work that have become particularly pioneering, important, guiding and popular in view of the development of pure mathematics in the second half of the 20th century. Also, it should be recollected that Chapters 4--7 of Bourbaki's ``Algebra'' focus on a systematic, rigorous and comprehensive treatment of field theory, Galois theory, and the theory of modules over a principal domain, together with numerous related topics and applications. More precisely, Chapter 4 develops the abstract theory of general polynomial rings, function fields, and formal power series, including the differential aspects (differentials and derivations) of these topics as a fundamental part. This is enriched by an in-depth treatment of symmetric tensor algebras, divided powers, polynomial maps, and their functorial interrelations, on the one hand, and by a just as comprehensive discussion of symmetric polynomials, symmetric rational functions, symmetric power series, resultants, and discriminants, on the other. Chapter 5 is then devoted to the theory of commuative fields, their various kinds of extensions, and the allied theory of étale algebras over a ground field. Apart from the fundamentals of Galois theory, Kummer theory, Artin-Schreier theory, and of the theory of finite fields, this chapter also discusses separable algebras and differential criteria for separability in full generality and detail. Chapter 6 briefly describes the basics of ordered groups and ordered fields, together with their respective fundamental structure theorems, whereas Chapter 7 deals with the theory of modules over a principal domain and its applications to the study of endomorphisms of finite-dimensional vector spaces. Today, fifty years after the first appearance of these chapters of Bourbaki's fundamental treatise on algebra, all their contents look pretty standard and can be found in various contemporary textbooks, too. Nevertheless, one has to keep in mind that this fact is due to the existence of Bourbaki's pioneering classic, which still serves as the prototype for the (ultra-)modern approach to abstract algebra, and as an indispensible reference for its numerous successors as well. In particular, it is the vast supply of Bourbaki-style exercises accompanying the entire text, together with the precious historical remarks to each chapter, that makes Bourbaki's text so absolutely unique and unsurpassed. No doubt, also Bourbaki's Chapters 4--7 of Book II will remain an everlasting treasure in the textbook literature of abstract algebra.
[Werner Kleinert (Berlin)]

MSC 2000:: *12-02 Research monographs (field theory)
13-02 Research monographs (commutative rings and algebras)
00A05 General mathematics
01A72 Schools of mathematics
12Fxx Field extensions
13F10 Principal ideal rings
13C10 Projective modules, etc.
12E05 Polynomials over general fields
06F15 Ordered groups
05E05 Symmetric functions

Keywords: general mathematics; textbook; field extensions; Galois theory; polynomial rings; principal domains; modules; algebras; endomorphisms; ordered fields

Citations: Zbl 0498.12001; Zbl 0719.12001

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	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.
	Elementary number theory. Primes, congruences, and secrets.

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