id: 06445028
dt: b
an: 2015f.00759
au: Lovett, Stephen
ti: Abstract algebra: structures and applications.
so: Boca Raton, FL: CRC Press (ISBN 978-1-4822-4890-6/hbk). xii, 708~p. (2016).
py: 2016
pu: Boca Raton, FL: CRC Press
la: EN
cc: H45 H75
ut: textbook (algebra); groups; rings; fields; modules; algebras; Galois
theory; computational algebra; categories; functors
ci:
li:
ab: The book under review provides a comprehensive introduction to the basic
concepts, principles, methods, results, and applications of modern
abstract algebra. As such it covers the standard material that is
usually taught in advanced algebra courses worldwide, however with
particular emphasis on the more general structural aspects underlying
the whole subject. This means, the book is not a treatise on general
algebraic systems and structures, but instead a presentation of the
core topics (groups, rings, and fields) following a consistent order
that leads the reader to other algebraic structures along the way. In
this sense, the term “algebraic structure” is used informally,
rather as a methodological organizing principle than a precisely
defined theoretical framework. In fact, it signifies that the special
order of presentation of the three standard structures follows one
unifying scheme: definition of structure, motivation, instructive
examples, general properties, subobjects, morphisms, important classes
and subclasses of the respective structures, quotient objects, actions
and action structures, related algebraic objects, and concrete
applications. Indeed, a second guiding principle of this text is to
illustrate how abstract algebra can be applied to various other
branches of mathematics, which is done by numerous examples, exercises,
so-called “investigative projects”, and a few extra sections
throughout the book. The latter include brief introductions to other,
related branches of algebra, thereby inviting the reader to further
study in these directions. As for the precise contents, the book
consists of thirteen chapters, each of which is organize in several
sections. Chapter 1 is of preparatory character and introduces the
necessary basic set theory, including sets and mappings, products,
operations, relations and equivalence relations, partial orders, and
Hasse diagrams. Chapter 2 presents the relevant basics from elementary
number theory such as the divisibility properties of integers, the
modular arithmetic of $\mathbb{Z}/n\mathbb{Z}$, and the principle of
mathematical induction. Chapter 3 is devoted to the elementary
properties of groups, with focus on subgroups, group homomorphisms,
group presentations, symmetric groups, and applications to cryptography
and geometry. Also, a first introduction to semigroups and monoids is
given as a structural generalization. Chapter 4 treats quotient groups,
including the isomorphism theorems and the fundamental theorem of
finitely generated abelian groups, while Chapter 5 discusses the
fundamentals of ring theory. Here rings generated by elements, matrix
rings, ring homomorphisms, ideals, quotient rings as well as maximal
ideals and prime ideals are the main objects of study. Chapter 6 turns
to the topic of divisibility in commutative rings, thereby covering
rings of fractions, Euclidean domains, unique factorization domains,
factorization properties of polynomials, rings of algebraic integers,
and the idea of RSA cryptography as a concrete application. Chapter 7
deals with fields and algebraic field extensions, together with
explanations of cyclotomic extensions, constructible numbers, splitting
fields, the algebraic closure of a field, and the basic properties of
finite fields. Chapter 8 returns to group theory by considering group
actions, Sylow’s theorems, and the idea of group representations,
whereas Chapter 9 discusses the classification of groups of given
(finite) order via composition series and solvable groups, finite
simple groups, semidirect products of groups, and nilpotent groups.
Chapter 10 is titled “Modules and algebras” and covers such basic
topics like Boolean algebras, free modules and module decomposition,
the structure theorem for finitely generated modules over principal
ideal domains, the Jordan normal form of linear transformations and its
applications, and a brief introduction to path algebras of directed
graphs. Elementary Galois theory is the theme of Chapter 11, in which
the following standard topics are touched upon: automorphisms of field
extensions, the fundamental theorem of Galois theory and its
applications, Galois groups of cyclotomic field extensions,
discriminants, computing Galois groups of polynomials, field of
characteristic $p>0$, and the problem of solving algebraic equations by
radicals. Chapter 12 describes some more specific topics in
computational commutative algebra, including introductions to
Noetherian rings, multivariable polynomial rings and their role in
affine geometry, Hilbert’s Nullstellensatz, monomial orders and
polynomial division, Gröbner bases, and Buchberger’s algorithm with
applications, respectively. This chapter ends with a brief introduction
to algebraic geometry via affine algebraic sets, the spectrum of a
ring, and their Zariski topologies. The final Chapter 13 gives a brief
introduction to categories and functors. This chapter is meant as a
precise formalization and generalization of the concept of “algebraic
structure”, which was emphasized (and loosely used) in the previous
parts of the book as a unifying methodological principle. The book ends
with two appendices on the algebra of complex numbers (A.1) and a list
of groups of order at most 24 up to isomorphism (A.2), respectively.
Also, there is a very carefully compiled list of notations, a rich
bibliography for references and further reading, and a just as helpful
index of notions. As already mentioned above, the utmost lucid,
detailed and versatile main text comes with a wealth of illustrating
examples and very instructive exercises in each single section of the
book, and each chapter ends with a section containing project ideas
(and hints) to challenge the student to write her or his own
investigative or expository papers on related topics. All in all, the
book under review is an excellent introduction to the principles of
abstract algebra for upper undergraduate and graduate students, and a
valuable source for instructors likewise. No doubt, this text is a
highly welcome addition to the already existing plethora of primers on
abstract algebra in the mathematical literature.
rv: Werner Kleinert (Berlin)