Abstract Algebra

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.