It is our conviction that the level of sophistication should increase, slowly at first, as the students become familiar with the subject. We think our ordering of the topics speaks directly to this assertion. In our classes we usually intend to cover Chapters 1, 2 and 3 in the first semester, and most of Chapters 4, 5 and 6 in the second semester.
The pace of a section per week allows time to assign and grade a significant number of problems, since in many ways this is a writing class. In practice, we usually begin the second semester with group homomorphisms and factor groups, and end with geometric constructions. The fourth edition has added a chapter and some harder problems to fill out this material. Separating the two hurdles of learning to write proofs and grasping abstract concepts makes the subject more accessible.
For example, cyclic groups are introduced in Chapter 1 in the context of number theory, and permutations are studied in Chapter 2, before abstract groups are introduced in Chapter 3. The ring of integers and rings of polynomials are covered before the general notion of a ring is introduced in Chapter 5.
The text emphasizes the historical connections to the solution of polynomial equations and to the theory of numbers. For strong classes, there is a complete treatment of Galois theory, and for honors students, there are optional sections on advanced number theory topics.
The first two chapters on the integers and functions contain full details, in addition to comments on techniques of proof. The intermediate chapters on groups, rings, and fields are written at a standard undergraduate level.
A free study guide is available online, with complete solutions to numerous supplementary problems. In the fourth edition we have added material to emphasize one of the key features of the book: the rising level of expectations as the students learn the subject. In the early chapters, we have added a few examples and exercises; in Chapters 7 and 8 we have added a significant number of more difficult exercises.
We have also added new material on groups in Chapter The third edition would probably not have been written without the impetus from George Bergman, of the University of California, Berkeley. Beachy is currently Professor Emeritus of Mathematical Sciences at Northern Illinois University, where he was a member of the faculty from to He has the title of Distinguished Teaching Professor awarded in He received his Ph.
He is continuing to do research in noncommutative algebra. Published by Waveland Press Co-authored with William D. It begins at a pace appropriate for students meeting serious proofs for the first time. As students mature, it progresses to a more demanding level. It typically introduces examples before developing the appropriate theories.
It contains a large number of exercises at a variety of levels.
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