This is a student-oriented text covering the standard first year graduate course in algebra. Solutions to all problems are included and some of the reasoning is informal.
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.
Table of Contents
- Group Fundamentals
- Ring Fundamentals
- Field Fundamentals
- Module Fundamentals
- Some Basic Techniques of Group Theory
- Galois Theory
- Introducing Algebraic Number Theory
- Introducing Algebraic Geometry
- Introducing Noncommutative Algebra
- Introducing Homological Algebra
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