What are the prerequisites for taking introductory abstract algebra?

Question

I am a maths student in my second year of university. I have taken and done quite well in Calculus I, II, III as well as a linear algebra (application focused) class. I have not worked much with proofs. My school's course catalog lists Abstract Algebra as one of the next courses but suggests a remedial "introduction to mathematical proofs" class for some. My question is if the community thinks it would be doable to go ahead with Abstract.

Our Abstract Algebra class is at the level of Thomas Hungerfords "Abstract Algebra: An Introduction".

Answer 1

If you have the interest, and are willing to work hard, go for it!

I first encountered proofs in linear algebra, and then in abstract algebra; it's a good domain of study for learning how to write proofs. I'd suggest you "skim" Hungerfords text in advance of the class to "preview" and become acquainted his style of writing and his manner of writing proofs..

At the same time you're previewing the course text, it might be wise to get a hold of the book:
How to Prove It: A Structured Approach by Daniel Velleman. I think you'd find it helpful to read and work through this book, at least in part, before taking the class. And in any case, it will serve as a good reference while taking the class, for help to better understand proofs and write them well.

How to Prove It... expands on each of the following topics:

*) The sentential (propositional) and predicate logic; quantificational logic
*) Set theory
*) Relations and functions
*) Mathematical induction and recursion
*) Infinite sets
*) Proof-writing

If you click on the link to the book, you can "preview" the book, and see the table of contents.

Other possible resources, both of which are highly regarded:

• How to Solve It, author G. Polya.
• Thinking Mathematically, authors J. Mason, L. Burton, K. Stacey.

Answer 2

You might want to look at the Proof books on my posting here: how to be good at proving?

You might also want to read this wonderful book as an introduction and truly work through it A Book of Abstract Algebra: Second Edition (Dover Books on Mathematics), Charles C Pinter (Author)

Lastly, I would recommend reading the responses here: A Book for abstract Algebra

Regards -A

Answer 3

You can take a taste of what it would be like with this free video lecture series given by a great teacher, Benedict Gross, at Harvard. He starts with the very basic principles and then builds. It's an outstanding presentation:

http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra

Added: You did mention that you studied linear algebra. But it sounds like your course was not rigorous. A theoretical linear algebra course - with theorems and proofs - is very helpful going into algebra and will also give you good math experience. So if you dept. offers such a course you might consider it.

Answer 4

If you've seen a bit of set theory (presumably in calculus) and done linear algebra - then you should be prepared for an introductory course in group theory.

The only way to develop skills with writing proofs is by experience. Abstract algebra is filled with enough interesting proofs to give you good examples for building intuition and plenty of challenging problems to keep you motivated.

You might consider seeing if there are any courses in the computer science department you could take concurrently, since it is typical for introductory level courses on data structures and algorithms to cover various proof techniques and strategies quite thoroughly. This is usually taught in conjunction with first order propositional logic and basic set theory.

If there isn't such a course, or if you cannot take one for some reason, then I recommend finding a copy of The Art of Computer Programming by Donald Knuth - which will help build the intuition needed for really thinking about proofs.

I recommend looking into computer science a bit because proving the correctness of an algorithm (and analyzing its complexity) requires the same level of rigor and employs the same strategies as the proofs in pure mathematics - but when proving an algorithm, you have something concrete to work with, whereas this is often not the case when proving a theorem.

Also, knowing how to prove the Euclidean algorithm and picking up some number theory from Knuth would certainly put you at an advantage in abstract algebra.

Answer 5

I would say there are no prerequisites at all!

Abstract algebra is one one hand a very self-contained subject. Everything can be defined abstractly, and you can prove interesting theorems without knowledge of anything else in mathematics. Knowing little bits of classical algebra, linear algebra, number theory and even calculus can help you to see some applications of what you learn, but studying the concepts by themselves does not require these things.

Something really nice about algebra is that you can study it in itself, but it also touches on nearly everything else in math in some way or another. Linear algebra is the prime example, but groups, rings and fields (the central objects of study in basic abstract algebra) are also really common in other areas. When you eventually go on to study other more specific topics, you will find these things crop up again and again, so it will be good to have seen them at least once before.

I don't know about the book that you'd be using, but in my experience, algebra is a fine place to start studying and proving theorems. In group theory, especially finite group theory, the basic theorems are all quite natural and accessible. As you delve into it, the material gets harder at a slow but steady rate, and the techniques and tricks used in the proofs also get more complicated. This is a good thing. It helps to broaden your mind and increase your sophistication.

As always, if you get stuck, don't be afraid to talk to your instructor, or ask here. I'm sure they'll be able to help you out. I'd like to throw out a few books that are good for studying algebra while you're at it, in order of my preference for them when I was at your stage.

1) Galian - Contemporary Abstract Algebra. Friendly. Concise, including only what you need to know for a first course. Basically everything you could want in a first book. At the end it has a bunch of interesting topics that you probably won't see in a first year course that are really cool to touch upon, since they are not any more advanced, just a little more niche.

2) Aluffi - Chapter 0. Has a completely heterodox viewpoint on algebra. Works from the categorical perspective. If you've not seen anything to do with categories before, this is probably harder. Nevertheless, the book is certainly a warm introduction to algebra and does not assume the reader knows any algebra.

3) Dummit & Foote - Abstract Algebra. I used this to study group theory on my own, and it contains lots of little gems in algebra. While it does not assume the reader knows any algebra, it is a massive book, and can be a little intimidating. The style is sometimes a little more terse. All in all though, persistence can be really rewarding. This book inspired my first "love" in mathematics in ring theory (specifically polynomial rings), and convinced me to do an independent study in polynomials of several variables and Grobner bases.