Abstract Algebra-The Basic Graduate Year (13).pdf

page 1 of Frontmatter
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 puter science as
well. In addition, I have attempted municate the intrinsic beauty of the subject.
Ideally, the reasoning underlying each step of a proof should pletely 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 substituting
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
representatio