id: 06530473
dt: b
an: 2016c.00792
au: Rotman, Joseph J.
ti: Advanced modern algebra. Part 1. 3rd edition.
so: Graduate Studies in Mathematics 165. Providence, RI: American Mathematical
Society (AMS) (ISBN 978-1-4704-1554-9/hbk). xiv, 706~p. (2015).
py: 2015
pu: Providence, RI: American Mathematical Society (AMS)
la: EN
cc: H45 H65 H75
ut: textbook (algebra); groups; rings; fields; Galois theory; modules;
categories and functors; linear and multilinear algebra; affine
algebraic geometry; Gröbner bases
ci: Zbl 1206.00007; Zbl 0997.00001
li:
ab: The first edition of J. Rotman’s comprehensive textbook “Advanced
modern algebra” was published in 2002 by Prentice Hall/Pearson
Education [Zbl 0997.00001], while the thoroughly revised second edition
appeared in 2010 within the renowned AMS textbook series “Graduate
Studies in Mathematics” [Zbl 1206.00007]. Now, after another five
years, there is a third edition of this excellent graduate algebra text
by the devoted teacher and author, Joseph J. Rotman, and again it comes
with significant changes and improvements. First of all, this new
edition is now divided into two separate volumes called Part 1 and Part
2, mainly for didactic reasons with respect to possible course
adoptions. In fact, these organizational changes are to reflect the
structure of graduate level algebra courses at the University of
Illinois at Urbana-Champaign, USA, with a wealth of additional material
incorporated to provide a wide range of options for course design
worldwide. The book under review is Part 1 of the new third edition,
and it consists of the following two basic courses: Galois theory
(Course 1) and Module theory (Course 2). As the author points out in
the preface, these two courses serve as joint prerequisites for the
forthcoming Part 2, in which more advanced topics in ring theory, group
theory, algebraic number theory, homological algebra, representation
theory, and algebraic geometry will be presented. As a result of the
reorganization of the material, the experienced textbook author J.
Rotman has rewritten many sections of the foregoing editions, and
thereby actually composed a completely new algebra book. As for the
precise contents of the present Part 1, there are 15 chapters divided
into the above mentioned two courses. Course I comprises the first
seven chapters covering the following topics. Chapter A-1 presents the
classical solution formulas for cubic and quartic polynomial equations,
while Chapter A-2 recalls some basics from (undergraduate) elementary
number theory such as divisibility, Euclidean algorithms, and linear
congruences. Chapter A-3 gives an introduction to the theory of
commutative rings and their ideals, including maximal ideals and prime
ideals, finite fields, irreducibility criteria for polynomials,
Euclidean rings, principal ideal domains, and unique factorization
domains. Chapter A-4 provides the first steps into group theory, ending
with the concept of simple groups, whereas Chapter A-5 treats
elementary Galois theory via Galois groups of extension fields,
solvability of algebraic equations by radicals, the fundamental theorem
of Galois theory, and the relevant aspects of group theory. At the end
of this chapter, concrete computations of Galois groups of polynomials
are presented. Chapters A-6 and A-7 are just two appendices reviewing
basic set theory and equivalence relations, on the one hand, and some
linear algebra on the other. Course II starts with Chapter B-1
introducing modules over noncommutative rings and chain conditions for
both rings and modules. This chapter concludes with diagrams and exact
sequences of module homomorphisms, together with the related technical
lemmas. Chapter B-2 is devoted to Zorn’s lemma and it various
fundamental applications to proofs of existence in algebra, including
algebraic closures, transcendence bases and Lüroth’s theorem.
Chapter B-3 turns to the applications of module theory to group theory
and linear algebra, with particular emphasis on the structure theorem
of finite abelian groups and canonical forms of matrices. This is
complemented by further topics in linear algebra: bilinear forms, inner
product spaces, and the classical linear transformation groups. Chapter
B-4 introduces categories and functors, with the special focus on
module categories. In this context, the reader meets here Galois theory
for infinite extensions, free and projective modules, injective
modules, divisible abelian groups, tensor products, adjoint
isomorphisms, and flat modules. Chapter B-5 is titled “Multilinear
algebra” and discusses the following topics: algebras and graded
algebras, tensor algebra, exterior algebra, Grassmann algebras,
exterior algebra and differential forms, determinantal calculus, and
related constructions in module theory. Chapter B-6 deals with more
advanced topics in commutative algebra. The first part of this chapter
explains the principles of the classical algebraic geometry of affine
varieties and their morphisms, including two proofs of Hilbert’s
Nullstellensatz, while the second part discusses some algorithmic
aspects of polynomials and the concept of Gröbner bases. Also Course
II ends with two appendices, Chapter B-7 introduces inverse limits and
direct limits in module categories, with an outlook to adjoint
functors, and Chapter B-8 reviews some general topology, with a brief
description of topological groups. Actually, these appendices
complement the section on Galois theory for infinite field extensions
in the previous Chapter B-4, where the respective notions were already
used. As in the foregoing editions, each section of the main text comes
with numerous related exercises, and the entire book is interspersed
with a wealth of illustrating, very instructive examples. In the
preface, the author expresses his hope that this new edition of his
standard algebra text presents the material in a more natural way,
making it simpler to digest and to use. Certainly, this will remain a
matter of taste among students and instructors, and the readers will
finally decide about that. Nevertheless, the highly experienced teacher
J. Rotman has again presented a new didactic arrangement of the
fundamental principles of abstract algebra, which is very original,
interesting, functional, inspiring, and student-friendly. No doubt,
also this rewritten third edition of the author’s classic “Advanced
modern algebra” will be among the most popular, useful and
appreciated textbooks in the field, and the mathematical community
should look forward to the appearance of Part 2 of this comprehensive
treatise in the not too far future.
rv: Werner Kleinert (Berlin)