abstract-algebrasoft-question
I am currently an undergraduate math student. (In fact, freshmen.)
I know that usually abstract algebra is taught somehow late in the undergraduate course, and curious how studies of abstract algebra at graduate level differ from studies at undergraduate level.
So, things like what gets new treatment, or what is learned new are what I want to know.
Here is what I tell my grad students:
The difference between undergrad mathematics and graduate mathematics is the difference between art history, or art appreciation, and learning to be an artist.
As an undergraduate you see a lot of mathematics, but you don't create new mathematics. The goal of graduate school (and here I am speaking from experience with top fifty U.S. graduate schools, so what I am saying probably applies best in that context) is to learn how to create new mathematics, and then to create that new mathematics.
One specific consequence of this (in my view) is the following: often in undergraduate mathematics classes, proofs and rigor are presented almost as moral imperatives --- as if it is a moral failing to know a statement without knowing why it is true; consequently, people often put a lot of effort into learning arguments just for the sake of having learnt them. (This is exaggerated, perhaps, but I think it reflects something real.) On the other hand, in research, one learns arguments for different reasons: to learn technique, to pick out important ideas --- there is a professional aspect to the way one looks at pieces of mathematics which is not usually present in undergraduate mathematics. One gives proofs in order to be sure that one hasn't blundered; one's interaction with the mathematics and the arguments is much more visceral than in undergraduate courses.
(I am not speaking from any experience now, but I think of the difference between learning how to interact with a block of marble, and bring a new form out of it, however rough it might be, in comparison to looking and learning about a lot of existing beautiful statues, masterpieces that they are.)
I will respond directly to this part of your question.
I would be happy if I could avoid such topics [analysis], but I don't know what type of mathematics is studied in a graduate degree level, so this leads me to the following questions:
What topics of Abstract Algebra should I study in depth ? what topics in Abstract Algebra should I be familiar with their basics ?
Are there any topics in analysis, topology etc' that are likely to be needed for answering a graduate degree level type of questions ?
What should be the focus of my work, should I try to do many exercises within the text, or focus on the proofs and the theory ?
Are there topics in Abstract Algebra, or other in other areas that I would need to know (maybe topology ?) that I can skip some parts of (mainly non-core topics that are hard to learn) since they would probably not help me (and due to lack of time) ? I have the book Abstract Algebra by Dummit and Foote to study with, as well as books in other area of mathematics such as Topology by Monkers that might help me with this goal.
Firstly, I want to mention that unless you are absolutely certain that you are going to specialize in pure group or ring theory, then you will need some analysis. In fact, you'll probably need a lot of analysis. Explaining why is a bit more complicated. The short version is that almost every area of math relies or is at least informed by analysis, algebra, and topology; this is why most graduate programs (in the US anyway) require these as either graduate classes or graduate entrance exams or graduate qualification tests, etc.
To expand in a slightly longer way - calculus is pretty interesting, and lets you do a lot of things. A common thing that mathematicians do is put measures on weirder spaces so that you can have some variant of integration. In number theory (even algebraic number theory, which is often the same thing as algebraic geometry, which is often the same thing as commutative algebra, which is just algebra and group theory), we really like having measures called Haar measures on matrix groups, like the $GL(n), SL(n), Symp(n)$, etc. This lets us do integration on these groups. So we study functions invariant under actions of these groups, or functions invariant on certain cosets of these groups that behave nicely under ring translation, or some similar idea. And one way we do this is to integrate them, or consider an integral over a weighted average of a function across the cosets our function is invariant over (read: Eisenstein Series for example), to extract largely algebraic information about number fields. Or we consider representations (as in representation theory, which I clump into the larger algebra domain sometimes) and analytic extensions of representations. Everything I've mentioned here requires a certain comfort with topology, analysis, and algebra.
This is to say that algebra mixes quite a bit with analysis in many ways. You would really benefit from having a good understanding of analysis and topology. In particular, don't focus solely on algebra. The other answer says this a little, but I am going to emphasize this a lot. It is very important to understand analysis and topology, unless you are going do limit yourself to pure, remote group theory. And even then, I wouldn't recommend it.
But back to your question at hand about algebra:
I would prescribe a path into algebra. In a comment on the other answer, you mention that you know groups, ring, fields, Galois theory. Cool! You also say you have Dummit and Foote (by far my preferred introduction to group and ring theory). Then I suggest two paths:
Go learn more about whatever parts you liked most. Sylow theorems interest you? Try to learn your way through Burnside's theorem. You like Galois theory? Pick up some infinite-dimensional Galois theory and try your hand. Maybe you already know that? Go pick up some algebraic number theory text - as an intro algebraic number theory text builds nicely on basic field theory and Galois theory, and suggests further paths. To be fair, I'm biased - I'm a number theorist. The important thing is that you go and dig deeper into things that interest you.
Pick up Atiyah and MacDonald's Commutative Algebra (hopefully from a library, as they're proud), and do your best at all the exercises. This is the 'natural' extension of what to do next, and it's the real path into a serious interest into algebra in my opinion. I say that you should do all the exercises because this book is famous for having really important lemmas and theorems in the exercises as opposed to the exposition. This will also really set your group theory and ring theory in stone, and you have Dummit and Foote to fall back on if you need. If you know this already, you should next go to Lang's Algebra (quite a bit, scary thing - take a look at it first), Matsumura's Commutative Ring Theory (much, much, much higher than Atiyah MacDonald, even though they have essentially the same name), or Eisenbud's Commutative Algebra (also harder than Atiyah MacDonald, but designed for people interested in algebraic geometry - if you don't know what that is, look it up).
I'd like to add one more thing about your (3) - the problem with learning the proofs and theory is that there is no reason for them to stick on their own. You might open up Atiyah MacDonald and understand everything you read, for example. But I wouldn't expect much of it to last, unless you use it. So a good general philosophy is to read and try to absorb, but then do exercises to let it solidify. Well written exercises require you to build on the text, both as a review and to build intuition.
A hard problem is knowing how many exercises to do. Too many, you waste your time. Too few, you'll forget much. But this is sort of moot, as it's hard to know what problems are useful or good to do before you actually do them, and in some texts some problems are much much better for you than others. For this, I advise you to ask your advisor (or find someone who can provide some sort of guidance) for direction once you have an idea what sort of things you want to learn about.
Best Answer
What constitutes a "graduate algebra' course in the United States has undergone a lengthy evolution. As MTurgon and rschwieb said earlier, it's highly subjective from teacher to teacher. But I think the evolution of the subject gives some insight of what can be expected at most programs today for a year long graduate course.
The first modern graduate text was,of course, Van Der Waerden's Moderne Algebra,based on the legendary lectures of Emil Artin, Emmy Noether and Van Der Waerden himself at the the University of Gottingen before World War II. It was the first real abstract algebra textbook since these lectures emerged from the researches of the authors.The syllabus became the standard mantra for algebra courses, "Groups,rings and fields". Until the 1960's, algebra was considered a graduate course and it was very unusual for most undergraduates to have had much exposure to algebra with the exception of the world's top programs, such as Harvard or Yale. The first undergraduate course in algebra was developed and presented by Saunders MacLane and Garrett Birkoff at Harvard in 1941 and it eventually became,of course, the basis for their classic text, A Survey of Modern Algebra. But it was very unusual for undergraduates to have a solid course in abstract algebra before the 1960's. When undergraduate and graduate algebra courses became standard courses in math departments, the curricula was fairly well-established: undergraduate courses were based on the Survey while graduate courses where based on Van Der Waerden. By the late 1960's and early 1970's, Van Der Waerden's book was no longer representative of the frontiers of algebra, which were now nearly unrecognizable with the explosive growth of categorical and homological methods,noncommutative algebra, modern commutative algebra and modern algebraic geometry. The first edition of Serge Lang's Algebra was published in 1965, concurrent with the peak popularity of the Bourbaki volumes. Lang's book effectively replaced Van der Waerden as the graduate algebra text of choice at top programs due its completely modern approach and it's emphasis on categorical and homological methods in all areas of algebra. It still is,to a large degree-but its sheer difficulty and dry austerity,coupled with the mammoth size of later editions and the explosively rapid growth of algebra at the research level-has recently lead to a new generation of algebra books at the graduate level, such as Grillet (my personal favorite), Rowen and Rotman.All these books have continued the hard categorical slant of Lang while trying to bring more recent developments into the standard courses.
From the prior discussion as well as my own experiences, I can state that graduate algebra generally differs from undergraduate courses in the subject in 3 ways:
1) Much the same way undergraduate analysis covers the "classical" analysis of the late 19th century and graduate analysis courses cover modern topic of the last century, graduate algebra differs mainly from undergraduate algebra in the emphasis on category theory and homological methods. There are programs that attempt to present category theory and diagram chasing to undergraduates in their algebra courses, but I think this is mainly at the top research programs,where the goal is to speed students to the frontiers as quickly as possible.In general, the categorical approach isn't tackled full on until the graduate algebra sequence and consequently the topics that are most strongly developed by these methods-i.e. homological algebra, noncommutative ring and module theory, algebraic geometry-are not discussed in depth until the graduate course.
2) Expect the course to be much deeper, terser and problem oriented then your undergraduate course. This for 2 reasons: a) A graduate course in algebra needs to survey most of the subject as it stands today to prepare the students for research in either algebra or other fields-and unless the student is ready to learn actively, there simply will not be time to cover the bulk of this work. Also b) graduate students are now beginning to make the transition to being professional mathematicians and they can't very well do that if they're still learning simple proofs off lectures or textbooks. They have to not only learn material much more quickly,they have to learn to build vast tracts of theory themselves. The best way to do both is to give the student a large chunk of the classwork to learn themselves.
3)Depending on whether your instructor is a prominent researcher in the field of algebra, a graduate algebra course may be much more closely tied to the frontiers of research then is usual. If so, he or she may cover the standard material in a "need to know" fashion in order to cover the maximal amount of material relevant to his or her research interests and a large chunk of the course would then be more like a research seminar, relying much more on published papers then standard textbooks. If the professor is not an active algebraicist, expect the course to follow a much more standard path through a conventional textbook like one of the ones stated above.
Specifically what topics can you expect in a graduate algebra class? At the minimum, I would expect the following topics to be covered: group theory through the Sylow theorems, free groups and presentations and the Fundamental Theorum of Abelian Groups, ring and module theory including both the noncommutative and commutative aspects, field theory including a large section on Galois theory, linear and multilinear algebra including a full discussion of tensor products,basic category theory and homological algebra,universal algebra, semisimple rings and algebras and perhaps some algebraic geometry and algebraic number theory.
Hope that helped-good luck!