Crash course in abstract algebra

[terhorst]

I'm thinking about taking the math GRE in December but I've never studied abstract algebra--all this about rings and groups just flies right over my head. Can anyone recommend a good introductory book? I'm thinking one of the Dover works might be good since they seem to emphasize problem solving, and the solutions are provided. And you can't argue with the price! Thanks.

 

[BSMSMSTMSPHD]

I just picked up https://www.amazon.com/dp/0486453561/?tag=pfamazon01-20 by Robert Ash to help me study for a PhD exam in Abstract Algebra. I really like this book - it's very concise and has tons of problems at the end of each section. Best of all, the answers are all explained in the back. The book is very comprehensive, covering groups, rings, modules and field theory, as well as a host of topics that I've never studied (categories, noncommutative and homological algebra etc.) I recommend it as a studying aid, but not as an encyclopedia of Algebra (think breadth rather than depth).

 

[mathwonk]

on my webpage there is a selection of free books on abstract algebra at different levels. and best of all, there are no answers to any of the exercises. in abstract algebra, if you need the answer, you have not understood the exercise, and often not the material it is based on either.

 

[Kummer]

Herstein?.

 

[mathwonk]

when i was a grad student studying from herstein, my experience was that it went in one ear and out the other. too slick? maybe? so i never really learned anything from it, except the problems were useful. but some students love it. i am talking abiout the original edition of topics in algebra.

and it is not a "crash course", i.e. not condensed like my phd prep notes. it is more elmentary however.

i also noticed sometimes that students who preferred herstein did not like my class so much. herstein tells you everything while i expect students to think more on their own.

so even though i read it as a student, it is one book i almost never refer to anymore, unlike the two artins, or lang, brauer, van der waerden, jacobson, Albert, hungerford, cartan-eilenberg, freyd, maclane, and occasionally dummit and foote.

about the only thing herstein has that other books do not do better in my opinion, is the structure theory of real orthogonal matrices, where instead of becoming diagonal, you have blocks of sines and cosines.

oh the last section, where he proves wedderburn's theorem on finite division rings, and lagrange's theorem on sums of 4 squares, is nice, but may have been omitted from later editions.

but again, the problems can be useful in preparing for a test.

 

[Kummer]

Beachy and Blair.

 

[mathwonk]

the reviews show that students mostly like b&b and the teacher thinks it is way too easy and slow, and more suited for high school. hence it seems appropriate for someone who has not had the subject and is teaching himself.

the crash course on my website for phd prep is way more sophisticated than the gre.

 

[Saladsamurai]

and best of all, there are no answers to any of the exercises. in abstract algebra, if you need the answer, you have not understood the exercise, and often not the material it is based on either.

Some of us employ visual learning into our studies and when teaching yourself a new subject, since you lack a teacher to lead by example, sometimes an explanation of a solution serves as the next best thing.

Peace, love and rainbows,

Casey

 

[Chris Hillman]

Read Topics, see... wonderful things

about the only thing herstein has that other books do not do better in my opinion, is the structure theory of real orthogonal matrices, where instead of becoming diagonal, you have blocks of sines and cosines.

Aw, c'mon, I think you're being too hard. But I agree that this decomposition is sadly neglected by authors of other textbooks (including Herstein's second algebra textbook, incidently).

oh the last section, where he proves wedderburn's theorem on finite division rings, and lagrange's theorem on sums of 4 squares, is nice, but may have been omitted from later editions.

What other undergraduate textbook covers the Hurwitz ring of integral quaternions? (For the OP, that's the analogue for quaternions of the Gaussian ring of integer complex numbers.) I've had occasion to actually use that stuff, at least in conversation with John Baez years ago on sci.physics.research.

 

[Chris Hillman]

And for those of us who do not need to boost our own egos, it is perfectly acceptable to use a book that explains the answers. mathwonk makes a generalization that all those who wish to be proficient in mathematics should adopt his learning method failing to realize that we are all a little different (or a lot).

I suspect mathwonk was thinking of the fact that huge chunks of modern algebra consist of theorems which are very important but which are also easily proven routinely from the definitions.

Another (patronizing?) thought occurs: anyone thinking of someday heading off to grad school should be grateful for prior experience at PF with insensitive comments, hurtful judgements, sweeping indictments, and broad sarcasm... My advice to those in this group is: acquire feathers like a duck and preen regularly

But I (and probably m too) agree that there is huge individual variation among modern algebra students at good unis. When I TAd such courses, I was struck by the fact that when I wrote up my solutions I invariably went for the most straightforward approach, but when I looked at what the students had written, some of them came up with quite different approaches. I remember one student in particular, an EMT taking the course for fun, who in my opinion was more talented than most of my graduate student peers. He always came up with highly original and delightful proofs, and I almost always added them to my solutions. Sometimes I had as many as four completely different solutions to the same problem!

Er...Casey, have I encountered you before here at PF under another handle?

 

[Saladsamurai]

I suspect mathwonk was thinking of the fact that huge chunks of modern algebra consist of theorems which are very important but which are also easily proven routinely from the definitions.

Another (patronizing?) thought occurs: anyone thinking of someday heading off to grad school should be grateful for prior experience at PF with insensitive comments, hurtful judgements, sweeping indictments, and broad sarcasm... My advice to those in this group is: acquire feathers like a duck and preen regularly

But I (and probably m too) agree that there is huge individual variation among modern algebra students at good unis. When I TAd such courses, I was struck by the fact that when I wrote up my solutions I invariably went for the most straightforward approach, but when I looked at what the students had written, some of them came up with quite different approaches. I remember one student in particular, an EMT taking the course for fun, who in my opinion was more talented than most of my graduate student peers. He always came up with highly original and delightful proofs, and I almost always added them to my solutions. Sometimes I had as many as four completely different solutions to the same problem!

Er...Casey, have I encountered you before here at PF under another handle?

No other handle. And I can appreciate the fact that not everyone should be treated with kid gloves and that it's good to become familiar with one's share of insensitive comments, hurtful judgements, sweeping indictments, and broad sarcasm but, I do not agree that PF is the place to do it.

Upom reflecting, my own comment seems a little harsh, so I have deleted it. Apologies to anyone who may have been offended.

Casey

 

[mathwonk]

notice herstein also has no answers to any exercises. if the exercises have answers you can never measure your progress. this is quite diffeent from worked examples which are very useful. notice my books have these in detail.

 

[tronter]

I like Dummitt/Foote the best. Tons of topics, and tons of exercises, good explantions.

 

[Kummer]

Nicalous Bourbaki on Algebra. (But I never read it).

Anything by Grothendieck has to be excellent.

 

[mathwonk]

grothendieck is very advanced. his article sur quelques points d'algebre homologique, was a watershed in the subject of sheaf cohomology, but i do not recommend it to young beginners. perhaps i am wrong to discourage anyone from reading this master, but i have noticed in my experience several beginners trying to read grothendieck, (including myself), early in their career and gaining little from it.

but if you do choose to read say his elements de geometrie algebrique, written in collaboration with dieudonne, i pass on mumford's advice: to read grothendieck, find the section you are interested in, read that, tracing back through all the references, then write it up yourself in two pages.

but it may be that the articles written by grothendieck himself, and not in collaboration with dieudonne, say on the construction of moduli, are more enlightening. some of them are really long though, and extremely challenging technically.

e.g. to study his riemann roch theorem, usually people recommend reading the article by borel and serre, rather than theorie des intersections.

so of course look at grothendieck's works, but probably dbne careful of immersing yourself for years unprofitably in one or two technical articles.

on the other hand, his famous unsuccssful grant application, "esquisse d'un programme", written at the end of his career, and recently translated in the book Geometric Galois actions by leila schnepps, is quite illuminating and has led to much beautiful work by others in the alst 20 years. and some of his general remarks, on doing research by maintaining the curiosity of a child, are very inspiring.

 

[mathwonk]

bourbaki's texts on algebra are extremely clear, and exist both in french and now english. they also have exercises and historical discussions.

 

[JonF]

the reviews show that students mostly like b&b and the teacher thinks it is way too easy and slow, and more suited for high school. hence it seems appropriate for someone who has not had the subject and is teaching himself.

the crash course on my website for phd prep is way more sophisticated than the gre.

What site are you talking about? Is it http://www.math.uga.edu/~roy/? I don’t see a book list there.

 

[mathwonk]

they are called course notes there. there are about 5 books, one 400 pages on grad alg, one 100 pages also on grad alg, one on undergrad alg, a 15 pager on linear alg, and one short note on the RRT.

 

[JonF]

thanks, i probably need to refresh my algebra and I'm low on funds so this works out well :)

 

[mathwonk]

there are other good free algebra books, maybe better than mine, by robert ash, and lee lady. you might google those.

also anything by james milne is just superb. see his website.