Learning Modern Algebra aligns with the CBMS
Mathematical Education of Teachers II recommendations, in both content
and practice. It emphasizes rings and fields over groups, and it makes
explicit connections between the ideas of abstract algebra and the
mathematics used by high school teachers. It provides opportunities
for prospective and practicing teachers to experience mathematics for
themselves, before the formalities are developed, and it is explicit
about the mathematical habits of mind that lie beneath the definitions
and theorems.
This book is designed for prospective and practicing
high school mathematics teachers, but it can serve as a text for
standard abstract algebra courses as well. The presentation is
organized historically: the Babylonians introduced Pythagorean triples
to teach the Pythagorean theorem; these were classified by Diophantus,
and eventually this led Fermat to conjecture his Last Theorem. The
text shows how much of modern algebra arose in attempts to prove this;
it also shows how other important themes in algebra arose from
questions related to teaching. Indeed, modern algebra is a very useful
tool for teachers, with deep connections to the actual content of high
school mathematics, as well as to the mathematics teachers use in
their profession that doesn't necessarily “end up on the
blackboard.”
The focus is on number theory, polynomials, and commutative
rings. Group theory is introduced near the end of the text to explain
why generalizations of the quadratic formula do not exist for
polynomials of high degree, allowing the reader to appreciate the more
general work of Galois and Abel on roots of polynomials. Results and
proofs are motivated with specific examples whenever possible, so that
abstractions emerge from concrete experience. Applications range from
the theory of repeating decimals to the use of imaginary quadratic
fields to construct problems with rational solutions. While such
applications are integrated throughout, each chapter also contains a
section giving explicit connections between the content of the chapter
and high school teaching.
This book covers abstract algebra from a historical perspective by using mathematics from attempts to prove Fermat's last theorem, as the title indicates. The target audience is high school mathematics teachers. However, typical undergraduate students will also derive great benefit by studying this text. The book is permeated with fascinating mathematical nuggets that are clearly explained
-- D. P. Turner, CHOICE
This book is destined for college students in the U.S. who intend to teach mathematics in high school. The reviewer finds it even more apt as a text for algebra courses. Special features in the book are side notes given and printed prominently at the margins of the pages, like: How to think about it, Historical notes, Etymology of notions and words. … The reviewer considers the book a refreshing read among the vast amount of books dealing with similar topics.
-- Robert W. van der Waall, Zentrallblatt MATH
Although this book is designed for college students who want to teach in high school," its mathematical richness fits it admirably as a text for a first abstract algebra course or a handbook for assiduous students working on their own. While definitions, examples, theorems and their proofs are organized formally, the book is enhanced by substantial historical notes, advice on "how to think about it," marginal comments, connections and etymology that are designed to "balance experience and formality." The book is tightly organized with the goal of elucidating developments leading to the solution of the Fermat conjecture and the theory of solvability by radicals.
-- E. J. Barbeau Mathematical Reviews
The primary intended audience of the book is future high school teachers. The authors take great pains to relate the material covered here to subjects that are taught in high school mathematics classes. … In writing this book, the authors have obviously kept the needs of the student reader firmly in mind at all times. The writing style is not just clear; iit is often conversational and humorous. … There are lots of exercises covering a wide range of difficulty, some with hints (but none with complete solutions) and there is a pretty good 39-entry bibliography. … What might be a very interesting use for this book would be as a text for a senior seminar or “topics” course for students who already have some prior exposure to abstract algebra. And, of course, whatever may be the applicability of this book as a text for undergraduate course, it seems clear to me that it belongs in any good undergraduate library.
-- Mark Hunacek MAA Reviews
This new edition, now in two parts, has been significantly
reorganized and many sections have been rewritten. The first part,
designed for a first year of graduate algebra, consists of two
courses: Galois theory and Module theory. Topics covered in the first
course are classical formulas for solutions of cubic and quartic
equations, classical number theory, commutative algebra, groups, and
Galois theory. Topics in the second course are Zorn's lemma, canonical
forms, inner product spaces, categories and limits, tensor products,
projective, injective, and flat modules, multilinear algebra, affine
varieties, and Gröbner bases.
The second part presents many topics mentioned in the first part in
greater depth and in more detail. The five chapters of the book are
devoted to group theory, representation theory, homological algebra,
categories, and commutative algebra, respectively. The book can be
used as a text for a second abstract algebra graduate course, as a
source of additional material to a first abstract algebra graduate
course, or for self-study.
Graduate students and researchers interested in learning and teaching algebra.
This book is the second part of the new edition of Advanced Modern Algebra (the first part published as Graduate Studies in Mathematics, Volume 165). Compared to the previous edition, the material has been significantly reorganized and many sections have been rewritten. The book presents many topics mentioned in the first part in greater depth and in more detail. The five chapters of the book are devoted to group theory, representation theory, homological algebra, categories, and commutative algebra, respectively. The book can be used as a text for a second abstract algebra graduate course, as a source of additional material to a first abstract algebra graduate course, or for self-study.
Graduate students and researchers interested in learning and teaching algebra.
This is a good book and a fitting tribute to a great mathematical expositor. I'll keep both the second edition and the two parts of the third edition on my shelf, and I'm sure I'll turn to them often in the future.
-- Fernando Q. Gouvêa, MAA Reviews
This new edition, now in two parts, has been significantly reorganized and many sections have been rewritten. This first part, designed for a first year of graduate algebra, consists of two courses: Galois theory and Module theory. Topics covered in the first course are classical formulas for solutions of cubic and quartic equations, classical number theory, commutative algebra, groups, and Galois theory. Topics in the second course are Zorn's lemma, canonical forms, inner product spaces, categories and limits, tensor products, projective, injective, and flat modules, multilinear algebra, affine varieties, and Gröbner bases.
Graduate students and researchers interested in learning and teaching algebra.
The organization of the text is clear and the pace is leisurely...recently I taught the second part of a first-year graduate course in algebra, and in the process dipped into this book at various points. I enjoyed the presentation, and the commentary on constructions and results aimed to place them in context, and I found the numerous examples and exercises helpful. I will be using this book when teaching such courses again, if not always as the main text, certainly as a useful supplement.
-- Srikanth B. Iyengar, Mathematical Reviews
Rotman is a wonderful expositor, and the two courses in this book strike me as well thought out and well presented.
-- Fernando Q. Gouvêa, MAA
This book is designed as a text for the first year of graduate algebra, but
it can also serve as a reference since it contains more advanced topics as
well. This second edition has a different organization than the first. It
begins with a discussion of the cubic and quartic equations, which leads
into permutations, group theory, and Galois theory (for finite
extensions; infinite Galois theory is discussed later in the book). The
study of groups continues with finite abelian groups (finitely generated
groups are discussed later, in the context of module theory), Sylow
theorems, simplicity of projective unimodular groups, free groups and
presentations, and the Nielsen–Schreier theorem (subgroups of
free groups are free).
The study of commutative rings continues with prime and maximal
ideals, unique factorization, noetherian rings, Zorn's lemma and
applications, varieties, and Gröbner bases. Next, noncommutative
rings and modules are discussed, treating tensor product, projective,
injective, and flat modules, categories, functors, and natural
transformations, categorical constructions (including direct and
inverse limits), and adjoint functors. Then follow group
representations: Wedderburn–Artin theorems, character theory,
theorems of Burnside and Frobenius, division rings, Brauer groups, and
abelian categories. Advanced linear algebra treats canonical forms for
matrices and the structure of modules over PIDs, followed by
multilinear algebra.
Homology is introduced, first for simplicial complexes, then as
derived functors, with applications to Ext, Tor, and cohomology of
groups, crossed products, and an introduction to algebraic
\(K\)-theory. Finally, the author treats localization, Dedekind
rings and algebraic number theory, and homological dimensions. The
book ends with the proof that regular local rings have unique
factorization.
[T]his is an excellent book containing much more than what is likely to be covered in a standard graduate course. It certainly fulfills the author's vision of a book that contains 'many of the standard theorems and definitions that users of Algebra need to know.' . . . Rotman has completely rewritten the book for the new edition. . . . The best features of the first edition are retained, including Rotman's humane and elegant approach to mathematical exposition: things are explained in both words and symbols, there are historical (and even autobiographical) remarks, and the etymology of some unusual terms is explored. Most importantly, the author often takes the time to put on paper the kind of 'here's how to think about it' advice that mathematicians often share with each other only orally. In the introduction, Rotman says that 'each generation should survey Algebra to make it serve the present time.' His Advanced Modern Algebra admirably fulfills that goal.
-- Fernando Q. Gouvêa, MAA Reviews
…a highly welcome enhancement to the existing textbook literature in the field of algebra.
-- Zentralblatt fur Mathematik