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Rating: 
Summary: Not exactly a rigorous text
Review: Although the material this book covers is important for any undergraduate mathematics student to learn, the approach Stahl takes is not as rigorous as a student planning to go onto grad school should get. If you are simply interested in a glimpse at group theory and Galois theory, then I recommend the book. If you want to really get your hands dirty, then I suggest possibly something by Herstein.
Rating: 
Summary: Hi.
Review: This year we used this text for an introductory course in algebra. I thought it did a very good job giving a motivation to the study of some more abstract-seeming topics (just like the plug on the back of the book says). I liked the way the book connects previous algebra a student might have taken with "modern algebra." I got along fairly well with this book because I read ahead a lot; however, many of my classmates had trouble with it. Stahl apparently does not always explain things fully. I have found myself at times being confused by his explanations of things that I already understood. One of my classmates told me he never would have made it through the last three months of the course if I had not explained things to him. I do not know how exactly the blame should be divided between Stahl and my classmates, but the experiment certainly did not work. After having taken a course with this book, I have some smaller pet peeves with which to deal as well. Nonstandard notation: Sometimes Stahl introduces nonstandard notation without explaining it as such. I helped explain things to people by saying, "Well, in S.S.'s notation, this means..." Order theorems: In Chapter 2, Stahl gives a list of theorems about orders of complex roots of unity. In Chapters 5, 7, 8, and 9, about the fields of integers modulo p, Galois fields, permutations, and abstract groups, respectively, he references back to those theorems proved in Chapter 2, saying "Clearly, these results carry over to..." "Clearly": Throughout the book, Stahl glosses over things by saying they are obvious. I think such "explanations" are a bad habit and can really confuse people. He leaves a lot to chance in hinging together the fundaments of mathematics by saying "clearly." Quaternions: In the chapter on abstract groups, he introduces the quaternion group without any explanation whatsoever as to what they actually are. He lists the elements as Id, A, B, C, D, E, F, G. All we are given to look at is a group multiplication table. "Huh, I wonder what this is for..." Thank god I already knew something about them. My teacher handed out some things about quaternions to the class too, because he too was amazed how little the book said about them. Together, I suppose we did Hamilton some justice. I also feel that this book is a little light on content. For my senior project, I have decided to read several books on algebra and Galois Theory and conduct a number of presentations for the benefit of myself and the members of my school's Math Department. When I began looking through Fraleigh, I was just overwhelmed with joy. It contained so many fascinating things... I was just delighted. I started at groups and just kept on reading. Algebra had so much in it I had never heard about. The Sylow Theorems. Principal Ideal Domains. Homomorphism. Direct products of groups, rings, etc. Ideals. The list just kept on growing. I was forced to conclude that the content in Stahl alone in no way would have prepared me for a second course in abstract algebra. Reading through the careful explanations in Fraleigh, I felt I had been cheated. Okay, maybe I got a little overemotional there. The point: do not read Stahl alone. Do look at Fraleigh. If you are looking for something a little less hardcore, look at Rotman (second edition), because most of the "important" things are laid out there in a warm yet still rigorous format. Stahl is great for the question of why we should care, but if you aim for a deep understanding of basic algebra, look elsewhere.
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