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Publication list for Rotman, Joseph

  • Learning Modern Algebra: From Early Attempts to Prove Fermat's Last Theorem
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    Title (HTML): Learning Modern Algebra: From Early Attempts to Prove Fermat’s Last Theorem
    Author(s) (Product display): Al Cuoco; Joseph J. Rotman
    Affiliation(s) (HTML):
    Imprint Blurb: MAA Press: An Imprint of the American Mathematical Society
    Abstract:

    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.

    Book Series Name: AMS/MAA Textbooks
    Volume: 23
    Publication Month and Year: 2013-12-31
    Copyright Year: 2013
    Page Count: 459
    Cover Type: Hardcover
    Print ISBN-13: 978-1-93951-201-7
    Online ISBN 13: 978-1-61444-612-5
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    Reviews:

    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

    Cover Image URL: ~~FreeAttachments/text-23-cov.jpg
  • Advanced Modern Algebra: Third Edition, Parts 1 and 2
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    Title (HTML): Advanced Modern Algebra: Third Edition, Parts 1 and 2
    Author(s) (Product display): Joseph J. Rotman
    Affiliation(s) (HTML): University of Illinois at Urbana-Champaign, Urbana, IL
    Abstract:

    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.

    Book Series Name: Graduate Studies in Mathematics
    Publication Month and Year: 2017-08-29
    Copyright Year: 2017
    Cover Type: Hardcover
    Print ISBN-13: 978-1-4704-4174-6
    Print ISSN: 1065-7339
    Primary MSC: 12; 13; 14; 15; 16; 18; 20
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    GSM/165
    GSM/180

    Readership:

    Graduate students and researchers interested in learning and teaching algebra.

    Cover Image URL: ~~FreeAttachments/gsm-165-180-cov.jpg
  • Advanced Modern Algebra: Third Edition, Part 2
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    Title (HTML): Advanced Modern Algebra: Third Edition, Part 2
    Author(s) (Product display): Joseph J. Rotman
    Affiliation(s) (HTML): University of Illinois at Urbana-Champaign, Urbana, IL
    Abstract:

    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.

    Book Series Name: Graduate Studies in Mathematics
    Volume: 180
    Publication Month and Year: 2017-08-15
    Copyright Year: 2017
    Page Count: 558
    Cover Type: Hardcover
    Print ISBN-13: 978-1-4704-2311-7
    Online ISBN 13: 978-1-4704-4170-8
    Print ISSN: 1065-7339
    Online ISSN: 1065-7339
    Primary MSC: 12; 13; 14; 15; 16; 18; 20
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    Supplemental Materials (URL at AMS): http://www.ams.org/bookpages/gsm-180
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    GSM/165/180

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    Graduate students and researchers interested in learning and teaching algebra.

    Reviews:

    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

    Related Products: Algebra Algebra: Chapter 0 Advanced Modern Algebra
    Cover Image URL: ~~FreeAttachments/gsm-180-cov.jpg
  • Advanced Modern Algebra: Third Edition, Part 1
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    Title (HTML): Advanced Modern Algebra: Third Edition, Part 1
    Author(s) (Product display): Joseph J. Rotman
    Affiliation(s) (HTML): University of Illinois at Urbana-Champaign, Urbana, IL
    Abstract:

    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.

    Book Series Name: Graduate Studies in Mathematics
    Volume: 165
    Publication Month and Year: 2015-11-30
    Copyright Year: 2015
    Page Count: 706
    Cover Type: Hardcover
    Print ISBN-13: 978-1-4704-1554-9
    Online ISBN 13: 978-1-4704-2731-3
    Print ISSN: 1065-7339
    Online ISSN: 1065-7339
    Primary MSC: 12; 13; 14; 15; 16; 18; 20
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    GSM/165/180

    Readership:

    Graduate students and researchers interested in learning and teaching algebra.

    Reviews:

    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

    Related Products: Algebra Algebra: Chapter 0
    Cover Image URL: ~~FreeAttachments/gsm-165-cov.jpg
  • Advanced Modern Algebra: Second Edition
    Base Product Code Keyword List: gsm; GSM; gsm/114; GSM/114; gsm-114; GSM-114
    Online Product Code: GSM/114.E
    Title (HTML): Advanced Modern Algebra: Second Edition
    Author(s) (Product display): Joseph J. Rotman
    Abstract:

    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.

    Book Series Name: Graduate Studies in Mathematics
    Volume: 114
    Publication Month and Year: 2010-08-11
    Copyright Year: 2010
    Page Count: 1008
    Online ISBN 13: 978-1-4704-1176-3
    Online ISSN: 1065-7339
    Primary MSC: 11; 12; 13; 15; 16; 18; 19; 20
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    Supplemental Materials (URL at AMS): http://www.ams.org/bookpages/gsm-114
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    Reviews:

    [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

    About the First Edition:

    …a highly welcome enhancement to the existing textbook literature in the field of algebra.

    -- Zentralblatt fur Mathematik

    Related Products: Algebra: Chapter 0 Algebra A Course in Algebra