| dc.description.abstract | This book is intended for students encountering the beautiful subject of abstract
algebra for the first time. My goal here is to provide a text that is suitable for you,
whether you plan to take only a single course in abstract algebra, or to carry on to
more advanced courses at the senior undergraduate and graduate levels. Naturally, I
wish to encourage you to study the subject further and to ensure that you are
prepared if you do so.
At many universities, including my own, abstract algebra is the first serious
proof-based course taken by mathematics majors. While it is quite possible to get
through, let us say, a course in calculus simply by memorizing a list of rules and
applying them correctly, without really understanding why anything works, such an
approach would be disastrous here. To be sure, you must carefully learn the definitions
and the statements of theorems, but that is nowhere near sufficient. In order
to master the material, you need to understand the proofs and then be able to prove
things yourself. This book contains hundreds of problems, and I cannot stress
strongly enough the need to solve as many of them as you can. Do not be discouraged
if you cannot get all of them! Some are very difficult. But try to figure out
as many as you can. You will only learn by getting your hands dirty.
As different universities have different sequences of courses, I am not assuming
any prerequisites beyond the high school level. Most of the material in Part I would
be covered in a typical course on discrete mathematics. Even if you have had such a
course, I urge you to read through it. In particular, you absolutely must understand
equivalence relations and equivalence classes thoroughly. (In my experience, many
students have trouble with these concepts.) From time to time, throughout Parts II
and III, some examples involving matrices or complex numbers appear. These can
be bypassed if you have not studied linear algebra or complex numbers, but in any
case, the material you need to know is not difficult and is discussed in the
appendices. In Part IV, it is necessary to know some linear algebra, but all of the
theorems used are proved in the text. | en_US |
