Abstract Algebra: The Basic Graduate Year

Author(s): Robert B. Ash, Departement of Mathematics, University of Illinois
Publication Date: October 2000

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Book excerpts:

This is a text for the basic graduate sequence in abstract algebra, offered by most universities. This text studies 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, this text attempts 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 the mission of this text to try to resolve the conflict.

The 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.
2. Brief informal surveys of large areas.
3. Intuitive arguments that replace lengthy formal proofs which do not reveal why a result is true.

Solutions to all problems are included in the text; most students will 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.

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