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Unformatted text preview: Abstract Algebra: The Basic Graduate Year Robert B. Ash PREFACE This is a text for the basic graduate sequence in abstract algebra, offered by most universities. We study fundamental algebraic structures, namely groups, rings, fields and modules, and maps between these structures. The techniques are used in many areas of mathematics, and there are applications to physics, engineering and computer science as well. In addition, I have attempted to communicate the intrinsic beauty of the subject. Ideally, the reasoning underlying each step of a proof should be completely clear, but the overall argument should be as brief as possible, allowing a sharp overview of the result. These two requirements are in opposition, and it is my job as expositor to try to resolve the conﬂict. My primary goal is to help the reader learn the subject, and there are times when informal or intuitive reasoning leads to greater understanding than a formal proof. In the text, there are three types of informal arguments: 1. The concrete or numerical example with all features of the general case. Here, the example indicates how the proof should go, and the formalization amounts to substi- tuting Greek letters for numbers. There is no essential loss of rigor in the informal version. 2. Brief informal surveys of large areas. There are two of these, p-adic numbers and group representation theory. References are given to books accessible to the beginning graduate student. 3. Intuitive arguments that replace lengthy formal proofs which do not reveal why a result is true. In this case, explicit references to a precise formalization are given. I am not saying that the formal proof should be avoided, just that the basic graduate year, where there are many pressing matters to cope with, may not be the appropriate place, especially when the result rather than the proof technique is used in applications. I would estimate that about 90 percent of the text is written in conventional style, and I hope that the book will be used as a classroom text as well as a supplementary reference. Solutions to all problems are included in the text; in my experience, most students find this to be a valuable feature. The writing style for the solutions is similar to that of the main text, and this allows for wider coverage as well as reinforcement of the basic ideas. Chapters 1–4 cover basic properties of groups, rings, fields and modules. The typi- cal student will have seen some but not all of this material in an undergraduate algebra course. [It should be possible to base an undergraduate course on Chapters 1–4, traversed at a suitable pace with detailed coverage of the exercises.] In Chapter 4, the fundamental structure theorems for finitely generated modules over a principal ideal domain are de- veloped concretely with the aid of the Smith normal form. Students will undoubtedly be comfortable with elementary row and column operations, and this will significantly aid the learning process....
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- Spring '11