I encourage people to write mathematical and cultural supplements to this book and to make them available on their websites. I do not wish to put any more topics into the book itself. If this book is to be useful as a base for other topics, it must remain brief and freely available. In this book there are no choices or options (except for what to add to it). You go straight ahead and you do everything. The path from the beginning of the book to the end of Chapter 5 is pretty much god-given. This is the core of algebra. However Chapter 5 (or maybe Chapter 6 for some) is as far as that holds true. After that, there is no god-given well-defined path forward. There are in fact, many good options, and people have to make decisions on which direction to take. Or you could be like Paul Revere, who got on his horse and rode off in all directions. Here are some possible topics.
- More classical finite dimensional linear algebra. Do the complex version of the last part of Ch 5. Do Hermitian, normal, and positive definite matrices, etc. (after Ch 5 or 6)
- Separable Hilbert Spaces. For students in science or analysis, this is a perfect continuation. Once in my algebra course I had several physics students, so I did Hilbert spaces after Ch 5, and they loved it. Well they should, because where else would it fit in so easily ? And it didn't hurt the math majors at all. (after Ch 5 or 6)
- More combinatorics and discrete mathematics for computer science majors. (anyplace it fits)
- Wedderburn theorem, group rings, Maschke's theorem, and group representations. I do this in more advanced courses, and it fits fine after the blue book. (after Ch 6)
- Commutative rings, localization, radicals, Nullstellensatz, spec, etc. (after Ch 6)
- More group theory. Sylow theorems, nilpotent groups, etc. (after Ch 5 or 6)
- Field extensions and Galois theory. (after Ch 5 or 6)
- Supplement of examples, problems, and exercises, to be used for homework assignments and exams. It is always easier to teach a course if you have copies of old exams. (throughout the book)
- Supplement using computers. Present examples and exercises using Maple, Mathematica, Matlab, or some other program. These could be symbolic or computational. (throughout the book)
- OTHER supplements that people select. The core of algebra is important in many subjects.
- Cultural and historical supplements. Some algebra texts have beautiful sections on the lives of famous mathematicians and on the history of mathematics. It would be pleasant and constructive to have cultural supplements to this text.
I hope this book will be useful, not only as a beginning textbook, but as a base of foundational abstract algebra on which other topics may be developed. We do not need to keep rewriting the core every year. The definition of group is not going to change, but the way mathematics is disseminated is going to change. It would be nice to develop a worldwide network of algebra texts. Ed C.