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

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 3

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 4

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.