Algebra: Chapter 0 (Graduate Studies in Mathematics)
Book Preface
This text presents an introduction to algebra suitable for upper-level undergraduate or beginning graduate courses. While there is a very extensive offering of textbooks at this level, in my experience teaching this material I have invariably felt the need for a self-contained text that would start `from zero’ (in the sense of not assuming that the reader has had substantial previous exposure to the subject) but that would impart from the very beginning a rather modern, categorically minded viewpoint and aim at reaching a good level of depth. Many textbooks in algebra brilliantly satisfy some, but not all, of these requirements. This book is my attempt at providing a working alternative.
There is a widespread perception that categories should be avoided at first blush, that the abstract language of categories should not be introduced until a student has toiled for a few semesters through example-driven illustrations of the nature of a subject like algebra. According to this viewpoint, categories are only tangentially relevant to the main topics covered in a beginning course, so they can simply be mentioned occasionally for the general edification of the reader, who will in time learn about them (by osmosis?). Paraphrasing a reviewer of a draft of the present text, `Discussions of categories at this level are the reason why God created appendices.’
It will be clear from a cursory glance at the table of contents that I think otherwise. In this text, categories are introduced on page 18, after a scant reminder of the basic language of naive set theory, for the main purpose of providing a context for universal properties. These are in turn evoked constantly as basic definitions are introduced. The word `universal’ appears at least 100 times in the first three chapters.
I believe that awareness of the categorical language, and especially some appreciation of universal properties, is particularly helpful in approaching a subject such as algebra `from the beginning’. The reader I have in mind is someone who has reached a certain level of mathematical maturity-for example, who needs no special assistance in grasping an induction argument-but may have only been exposed to algebra in a very cursory manner. My experience is that many upper-level undergraduates or beginning graduate students at Florida State University and at comparable institutions fit this description. For these students, seeing the many introductoiy concepts in algebra as instances of a few powerful ideas (encapsulated in suitable universal properties) helps to build a comforting unifying context for these notions. The amount of categorical language needed for this catalyzing function is very limited; for example, fuuctors are not really necessary in this acclimatizing stage.
Thus, in my mind the benefit of this approach is precisely that it helps a true beginner, if it is applied with due care. This is my experience in the classroom, and it is the main characteristic feature of this text. The very little categorical language introduced at the outset informs the first part of the book, introducing in general terms groups, rings, and modules. This is followed by a (rather traditional) treatment of standard topics such as Sylow theorems, unique factorization, elementary linear algebra, and field theory. The last third of the book wades into somewhat deeper waters, dealing with tensor products and How (including a first introduction to Tor and Ext) and including a final chapter devoted to homological algebra. Some familiarity with categorical language appears indispensable to me in order to appreciate this latter material, and this is hopefully uncontroversial. Having developed a feel for this language in the earlier parts of the book, students find the transition into these more advanced topics particularly smooth.
A first version of this book was essentially a careful transcript of my lectures in a run of the (three-semester) algebra sequence at FSU. The chapter on homological algebra was added at the instigation of Ed Dunne, as were a very substantial number of the exercises. The main body of the text has remained very close to the original `transcript’ version; I have resisted the temptation of expanding the material when revising it for publication. I believe that an effective introductory textbook (this is Chapter 0, after all…) should be realistic: it must be possible to cover in class what is covered in the book. Otherwise, the book veers into the `reference’ category; I never meant to write a reference book in algebra, and it would be futile (of me) to try to ameliorate excellent available references such as Lang’s `Algebra’. The problem sets will give an opportunity to a teacher, or any motivated reader, to get quite a bit beyond what is covered in the main text. To guide in the choice of exercises, I have marked with a c> those problems that are directly referenced from the text, and with a -l those problems that are referenced from other problems. A minimalist teacher may simply assign all and only the t> problems; these do nothing more than anchor the understanding by practice and may be all that a student can realistically he expected to work out while juggling TA duties and two or three other courses of similar intensity as this one. The main body of the text, together with these exercises, forms a self-contained presentation of essential material. The other exercises, and especially the threads traced by those marked with will offer the opportunity to cover other topics, which some may well consider just as essential: the modular group, quaternions, nilpotent groups, Artinian rings, the Jacobson radical, localization, Lagrange’s theorem on four squares, projective space and Grassmannians, Nakayama’s lemma, associated primes, the spectral theorem for normal operators, etc., are some examples of topics that make their appearance in the exercises. Often a topic is presented over the course of several exercises, placed in appropriate sections of the book. For example, `Wedderburn’s little theorem’ is mentioned in Remark 111.1.16 (that is: Remark 1.16 in Chapter III); particular cases are presented in Exercises 111.2.11 and IV.2.17, and the reader eventually obtains a proof in Exercise VII.5.14, following preliminaries given in Exercises VII.5.12 and VII.5.13. The – label and perusal of the index should facilitate the navigation of such topics. To help further in this process, I have decorated every exercise with a list (added in square brackets) of the places in the book that refer to it. For example, an instructor evaluating whether to assign Exercise V.2.25 will be immediately aware that this exercise is quoted in Exercise VII.5.18, proving a particular case of Dirchlet’s theorem on primes in arithmetic progressions, and that this will in turn be quoted in §V11.7.6, discussing the realization of abelian groups as Galois groups over Q.
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