# Groups and Rings course suggestion

I am a physics undergrad, looking to explore pure maths.

I apologize if this question is not appropriate for MathSE, but I couldn't resist posting it. Feel free to close it down.

I haven't taken any formal course in maths uptil now, but have done some linear algebra, representation theory, topology, differential geometry by self-study from mathematical physics textbook. I am thinking of taking a masters level course on groups and rings at my uni. The course structure is here.

http://mat.iitm.ac.in/msc%20course%20content/ma532.html

Please could you guide me if it would be wise to directly take a MSc level algebra course with no prior exposure to a formal pure math course. This may sound immature, but I can physically put only maybe 7-8 hours a week outside class for this course. I have an ongoing research project, 6 theory courses, and independent QFT. I am really interested in learning this, but I have no idea if it would be too ambitious.

In particular my question is :

1. How many hours of effort would I require to put for this course (check the link)?

2. Would I able to cope up with rigorous, and problems. I am really not aware of the rigour in mathematics, but i really enjoy how simple definitions can lead to beautiful results, and help to express things clearly.

• btw if you end up taking the course, consider checking out Herstein (as mentioned on the website) - I found it super helpful when I was learning this stuff! – uncookedfalcon Jan 7 '13 at 18:13
• Whats the difference between his too books, 'Topics in Algebra' and 'Abstract Algebra'? Which of them are you recommending? – user23238 Jan 8 '13 at 3:50
• oh yeah a fair point - I was recommending 'Topics in Algebra', and haven't looked at the other – uncookedfalcon Jan 8 '13 at 6:38

I may be going against the grain here, but this is a bad idea.

If you have never taken a formal mathematics course you probably aren't going to be familiar enough with proof writing to attempt a second year algebra course. It's talking about review of groups and rings, which means you should probably have worked with them before. It would be one thing if you could put a lot of time in it, but only $5$ to $6$ hours a week? I've had single problems take me that long in abstract algebra courses. If you're really taking $6$ other courses and doing research there's no way you're going from zero to master's level algebra.

• Thanks a lot for the reply. This seems to be the popular advice given by by peers and friends. I just wanted to be sure, by asking the Math SE community. I don't want to regret later not taking this course, and I am pretty sure I won't be able to take this course later in the future during undergrad as the slot is usually taken up. – user23238 Jan 7 '13 at 18:56
• @Alexander : To go against your answer, group and ring theory is not really master's level in algebra. I think the reason why they call it "review" is because they're going to skip some stuff in order to combine a year of material in one semester. Notice how there is nothing else but "reviews" in the course content. I agree he should not take the course for his sanity (as said in my answer). The rest of your answer is totally relevant. +1 – Patrick Da Silva Jan 7 '13 at 19:36
• @ramanujan_dirac Believe me, it hurts to recommend that somebody not take a course in my favorite field of mathematics- it would be a shame to miss the opportunity to take group and ring theory as they are both beautiful subjects. Maybe you could audit, or just sit in on lectures? – Alexander Gruber Jan 7 '13 at 22:59
• I have decided to take the course. :) – user23238 Jan 9 '13 at 16:45
• Would you recommend Dummit&Foote, Herstein or Artin? I foind Artin the best and most logically written. – user23238 Jan 9 '13 at 16:46

The course you are taking, by the contents described on your link, would be the two algebra courses given in my pure math program for undergrads combined together (one concerning group theory, the other one concerning ring theory). If you are thinking about doing this in one semester, I suggest a little maturity in algebra, and linear algebra is definitely not enough.

I know some physicists myself who did a math-representation theory course after doing a lot of physics (including the study of characters as they do it in physics courses), and their first reaction was that "they had never seen this before". In other words, they had studied characters the way it's done in physics, which is totally taught differently than the way it's done in math for the simple reason physicists use characters, but mathematicians prove statements about them, which is a different point of view.

You could also ask your teacher about it ; if the course is targeted at mathematicians, I would say you would spend a huge amount of time on it and wouldn't suggest you do that. If the course is targeted at physicists, then you can have a shot.

If you have never done "proof-style math" (which for mathematicians is just "math"), then don't even think about taking the course, you will spend too much time on it(you could be able to do it but you would pay the price). You can still use the reference they give to read about it though (Dummit & Foote) as it is an excellent book, or ask the teacher if you can just sit in the course and listen, do the exercises at home when you can and show them to your teacher if he's friendly enough to help you learn.

Hope that helps,

• Thanks, this is a really well thought answer, as you know how differently physicists and mathematicians do math.This answer really helps. Though, the (irrational) part of me wanting to do the course, is extremely stubborn at this moment. I would be great if I get some more illuminating answers. – user23238 Jan 7 '13 at 18:42
• @ramanujan : I would correct your comment by saying that physicists don't do math, they do physics. I never saw a physicist prove a theorem in a physics course... (in the sense that they don't do proofs). Most of the time they make sense out of things by computing stuff and by giving the main idea. – Patrick Da Silva Jan 7 '13 at 19:33
• Yeah. I agree with that. – user23238 Jan 7 '13 at 19:50

I agree with Jasper's comment regarding the relative difficulty of subject material.

However since, to your credit, you recognize the benefits of becoming more skillful in proofs, perhaps I could make the following suggestion.

It might be beneficial for a period of time cultivating that skill. It will serve you immensely in the future. So think of it as training. Toward that end, here are two suggestions for self-study (pick one) that will be very useful:

-- This is a link to a download of a real analysis course taught by Fields Medal winner Vaughan Jones. He is a master and presents that material so as to cultivate your ability do do proofs. Real analysis is a key element in math study and is important if you want to go on to functional analysis: