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“Undergraduate Algebra”: Or How I Relearned Algebra in a Week *August 11, 2009*

*Posted by Martin Camacho in General.*

Tags: algebra, books

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Tags: algebra, books

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A few weeks ago I vowed to relearn all of my forgotten algebra – advanced group theory, rings, modules, and fields especially. The main problem,at least for me, was finding a viable resource to tutor me. Wikipedia proved futile as there was no use in clicking links in an unsystematic manner, and Wikibooks’ algebra section was simultaneously obtrusive and incomplete.

Akhil Mathew had recommended Lang’s ‘Algebra’, part of Springer’s series of Graduate Texts in Mathematics, but I found it much to technical for a refresher in basic undergraduate algebra. One of my advisers and I had worked from Dummit and Foote but I found it much more referential and dense than an introductory text. I then realized that I had been examining graduate-level books which were much more concerned with advanced algebra – commutative and non-commutative rings, Noetherian rings, etc….

Browsing through the Princeton bookstore’s mathematics section I stumbled across Lang’s “Undergraduate Algebra”, and subsequently decided that I’d buy it. After a week I feel much more comfortable with groups, rings, and fields. Lang’s expository style is much simpler than his style in “Algebra”, and I recommend ‘Undergraduate Algebra’ with full praise!

Artin Artin Artin Artin

Also: I somehow doubt you’re old enough to have “forgotten” algebra. Yesterday my officemate got asked in his dissertation defense what the difference is between a ring and a field. He didn’t remember, and when he asked me, I was like, “well, the difference between a ring and a group …” so we had to look up fields. It was embarrassing. But then, we’re in astronomy, and not math, so there’s that…

Don’t get a copy of the first edition; Artin’s releasing a second edition sometime soon and it’s quite good. (In particular, there seem to be a lot more exercises.)

I didn’t realize you were looking for a quick brush-up on undergraduate algebra, or I definitely wouldn’t have recommended Lang’s Algebra! In my opinion, it is a very good textbook, with a lot of interesting material, but it’s sometimes written rather carelessly (presumably because Lang wrote a lot of stuff), and lacks motivation. The topics are not always covered as thoroughly as they could: for instance, his introduction to commutative algebra seems to me too bare-bones, especially since he says that he is trying to include material for Hartshorne; it would have been much more relevant to discuss, say, dimension theory instead of elimination theory or real fields. In a sense, he tries to do a bit of everything, and that’s just not possible in 900 pages. I myself was unable to learn material from several chapters, and I only understood them (sometimes) after I read some of the same things elsewhere.

That said, it’s more fun to read than most other algebra textbooks I have seen, partially because Lang tends to emphasize category theory more than most people.

At least I didn’t recommend Bourbaki though…

For what you wanted I think my favorite book is Allan Clark’s Elements of Abstract Algebra.

I discovered it following my own meta-advice on math books: go to the library and pick the thinnest book on the subject you’re interested in. (Of course, what I really mean is something like browse through all the books available and pick the one that suits you best, but start by and pay more attention to the thinnest ones. I often do wind up reading the thinnest book I can find, though.)

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