Artin Algebra Solutions Chapter 1

[Ludivina Speed]

This section contains solutions to selected problems from Artin'sAlgebra, 2nd ed. I have skipped most of the chapters, focusing onfinite groups, rings, and fields. I have also skipped certainsections that I didn't find interesting or useful, as well as certainproblems (especially those that seemed too tedious).

I will probably not be solving any more problems in this book, but you mayfeel free to email me with errata, or to contribute better solutions thatcan be understood using material covered by the appropriate point in thetext.

So far the exercises are not particularly challenging. This worries me because I'm not a student at any university and I have no mathematician friends to talk to and so the only way for me to check my level of understanding is doing the exercises.

All in all, I think the book can be a really good place to learn algebra. Obviously this is only my personal opinion, there will certainly be others (probably knowing much more than myself on the subject) with different views on the subject.

Algebraic arguments are generally more elegant, and more enlightening, when one works with arrows rather than elements. Don't worry about the book not being "serious" enough; the whole point of the text is to start you off thinking with the same language as "serious" mathematicians in algebra-heavy disciplines. The main reason to stay away from Aluffi is that category theory is rather abstract, and can seem difficult and/or pointless until one has built up a library of examples. This doesn't seem to be a problem for you, so Aluffi is probably a pretty good choice.

Saal, I second the opinion, based on starting the book, that Aluffi has one of the most user-friendly intros to category theory around. And that is a very good thing because one can relatively easily make one's way through the basics of abstract algebra then hit the wall of categorical thinking and get lost and discouraged. And given its importance as well, it is very nice that you seem to be getting it.

As others as pointed out, if you want more on group representations, you can look at Dummit and Foote, or Lang, and I'm sure Jacobson's Basic Algebra (yes, it has long chapters on Galois Theory in vol. 1 and Rep. Theory in vol. 2). Also, Emil Artin's little book on Galois Theory I recall as being very concise and clear.

Indeed, if you want something that is a bit more 'concrete,' and less concerned with arrows and diagrams, then in increasing order of difficulty, these are excellent books. I am currently concentrating on Dummit and Foote while starting to look at Aluffi for decent insight:

Last semester I picked up an algebra course at my university, which unfortunately was scheduled during my exams of my major (I'm a computer science major). So I had to self study the material, however, the self written syllabus was not self study friendly (good syllabus overall though).

So, now I want to ask whether any of you know any good books on abstract algebra, which lift off at basic ring theory and continue to more advanced ring theory and to finite fields, Galois theory, ...

There's always the classic Abstract Algebra by Dummit and Foote. Section II of the text gives a nice treatment of ring theory, certainly providing plenty of review for what you have already covered while introducing more advanced concepts of ring theory. Section III will cover the field and Galois theory you're interested in. Some of the exercises can be difficult at times, especially for self-study, but the authors tend to give a number of examples and always provide the motivation for why they are doing what they are doing.

"Contemporary Abstract Algebra 7/e provides a solid introduction to the traditional topics in abstract algebra while conveying to students that it is a contemporary subject used daily by working mathematicians, computer scientists, physicists, and chemists. The text includes numerous figures, tables, photographs, charts, biographies, computer exercises, and suggested readings giving the subject a current feel which makes the content interesting and relevant for students."

It's a unique book that covers the basics of group theory, ring theory, and even a tiny bit of Galois Theory, but it does it almost entirely through problems. Every chapter begins with a short section defining some terms and giving a few basic proofs, and then it leads the reader through the rest of the exposition in a series of problems, some difficult, some not. The end result is that if you actually do all the problems, you've written the book yourself. It's impossible not to be comfortable with basic abstract algebra if you take this book seriously.

One book that I did not see mentioned, but which really deserves some accolades is the recent book Visual Group Theory by Nathan Carter. There are some excellent accompanying videos by Prof. Macauley on his youtube channel. These go really well together.

The biggest trouble I ran into with group theory and abstract algebra was the dizzying set of definitions that most books present at the beginning. You get a bunch of definitions with little or no motivation and with little description of the underlying geometry of how the binary operations work. The nice thing about the Carter book and the videos is that it spends a lot of time working though group diagrams and showing the "symmetry" of a group. It is easy to get caught up in the formalism, but without a good intuitive understanding of how different groups work--and how simple groups differ--it is easy to get frustrated--especially in self study. Also note that the Carter book has exercise solutions at the end.

There is no easy or right answer. I know brilliant professors who cannot easily decide what textbook to use for an advanced math course, and for good reason. Every book has its own strengths and weaknesses. I suggest you go to your math library (assuming one is available during this pandemic) and examine several books. A book you like might be hated by someone else, it is highly individual. You likely will need at least two or three books so you can go back and forth. Even a good book can be bad in a particular section and vice versa. Use a common textbook that has gone through at last two or three editions as a guide as to what topics to cover and then be prepared to use alternate books to actually learn the topic.

Use the Internet. Don't be afraid to read lecture notes or check Wikipedia. Also, Professor Keith Conrad (Univ. of Conn.) has dozens of expository papers on algebra on his web site, some are easy, some are difficult, and some are advanced or specialized. I have found that lectures by professors at lessor known universities to often be better than those by professors at famous brand name universities. That being said, I have found lectures by Unv. of Berkeley professors to be quite good, and lectures by MIT professors to also be good, but the latter are often very fast paced and better for review than to self learn from as they are so intense.

I suggest you get an easy book, an intermediate book and eventually a hard book. Herstein's: Topics in Algebra is harder than Birkhoff and MacLane's book, but Birkhoff and MacLane's book is good for learning the fundamentals. As an undergraduate I used Herstein, but I think it is too difficult to self study from.

It is critical to learn the definitions and other fundamentals cold and then go on to a more advanced treatment. (One really smart professor basically told me: memorize definitions, but do not memorize proofs, just understand them.) Herstein loves to give problems and results that are hard using elementary methods, but easy using more advanced methods. In my opinion this a bad way to learn, as not everybody is clever at solving hard problems or following highly technical arguments, and I think it is more useful to put one's energy into learning the concepts and theory that makes it possible to eventually easily understand what is really going on, rather than rely on clever technical tricks or manipulations to get a result with no real deep understanding as to what is really going on. Neither Herstein nor Birkhoff and MacLane cover everything a graduate course would cover. Herstein, in my opinion, makes the subject seem more difficult than it is.

I also use: Algebra: A Graduate Course by I. Martin Isaacs, it has its strengths and weaknesses. It is is elegant and the proofs are carefully done, but it may be too abstract and condensed to self study from.

A classic is the two volume (mostly of the time only the first volume is used) set by B. L. van der Waerden titled: Modern Algebra. There is at least two English editions (the original is in German). Even though the editions differ, any English edition is fine. And if you really groove on abstraction there is Serge Lang's book, simply titled: Algebra.

Good luck. I think it is great you are so motivated. Keep the faith. Don't worry if at times you get overwhelmed or discouraged --- self study is not easy --- it has happened to many of us at some point in time, yet somehow we didn't let it stop us, nor should you.

I am speaking from the standpoint of a student, and I think that a very good book on introductory abstract algebra that doesn't get mentioned very often is Basic Algebra by Anthony W. Knapp. From experience, the text is accessible with very little pre-requisite knowledge, is less "talkative" than Dummit and Foote (and in my opinion, definitely not dry, unlike D&F), and more rigorous in exposition than Artin's Algebra, although Artin's book is a good and standard first text as well.

Knapp covers most basic topics that the undergraduate student needs to know and is largely self-contained. I think, for the first seven chapters of this book, you can't really do much better by way of alternative texts. However, you could supplement or even replace the eighth chapter with Introduction to Commutative Algebra by Michael Atiyah and Ian MacDonald. However, if you are reading algebra for the first time, I don't suggest using Atiyah's book, unless you are feeling very confident or very lucky! :) Having said that, it is an excellent book and you should try reading it at some point. For the ninth chapter, you could use Emil Artin's classic little book on Galois Theory, based on his lectures on the subject. Another good reference which I haven't used but heard quite a few good things about is Nathanson's Basic Algebra: I (Chapter 4 (?), I think). Yet another book on Galois Theory is D.J.H. Garling's Galois Theory, which is where I initially learnt my Galois Theory from. As for Chapter 10 in Knapp, I have nothing to say, since I never got down to reading it.

c80f0f1006