Abstract Algebra Seems like most of the math majors at my school call themselves algebraist. I really am unsure what an algebraist does. It seems like they're the mathematical equivalent of biologists, observing, categorizing, all the while linking categorizations and members thereof together in new (sometimes surprising ways). But having a name and label for everything (it's been done with finite groups I believe) seems to uninteresting a goal for so many people to be algebraist. Indeed, over categorization and labeling breeds repugnant amounts of technical terms. I know some people like to namedrop with technical terms, but to me it seems more beneficial working to not use the technical terms, to be able to explain to those without the background. Even our major tools, the morphisms, are just ways of categorizing new groups/rings into a variety of already encountered sub types of rings/groups. Such a goal would be wholly useless for someone doing analysis on differential equations. 
Re: Abstract Algebra In the grand scheme of things, what mathematicians do is categorize and describe increasingly abstract structures. This isn't a pursuit unique to algebraists. The Poincare conjecture was part of a classification movement which is similar in spirit to the classification of finite groups: What manifolds are diffeomorphic to R^n? To S^n? Through the 20th century, you saw the same push for this classification as you did for FSGs. You also see similar attempts to classify things in graph theory there are two "forbidden" minors for planarity, but there are 33 (I think?) for the projective plane, and hundreds for other spaces; graph theorists are spending a good deal of time categorizing embeddability. When you look at category theorists (who I consider to be algebraists...) you see that they aren't categorizing (uh... sorry) anything any more than anyone else in fact higher dimensional category theorists are just starting to really figure out what exactly it is they're trying to talk about; they don't have a whole lot of time to worry about how to taxonomize these things. Regarding termdropping: Think of terms like Hausdorff, regular, normal, and compact in topology (and continuous, uniformly continuous in analysis); these are all convenient shorthands that say "the object we are looking at satisfies some extra properties." These extra properties give us information about the structure we are looking at. Would you really rather I say "Let G be a topological space for which every open cover has a finite subcover" every time I talk about compact spaces, or would you rather I say "Let G be compact" and move on to what I'm trying to say? Yes, there are people who like to drop big words to feel good about themselves, but the point of these abstract, esoteric definitions is not to be esoteric or precocious the point is to get past the things we see over and over again, and move onto what we're trying to talk about. The words, like any word, are a way for us to communicate information efficiently. Because mathematicians work with new structures all the time, we have to also be in the business of creating language. Since we are trying to describe structures for which there has never been a need for words, by using other structures for which there has never been a need for words, anything we tried to say would very quickly become unruly if we didn't have a quick way of saying it. My favorite example recently is from ETCS (a structuralist set theory): the axioms for it can be very conveniently stated "The category of sets is a wellpointed topos with a natural number object satisfying the axiom of choice." If you know what a wellpointed topos is, what a natural number object is, and what the axiom of choice is, then you understand the axiom system. Compare that to ZFC while the ZFC axioms might be easier to pick apart (explaining the whole axiom system for ETCS in words that "any" mathematician could understand immediately would take... a while), a number of mathematicians are familiar with all 3 of the things needed to understand that axiom (at least, as familiar as they are with the formalism of ZFC) from other areas, so this sentence conveys a good deal of information so long as you have the language. It allows someone talking about ETCS to move past the definition, and get to "real" mathematics quicker. Anyway, onto your question "what does an algebraist do?" That's a difficult question, in large part because "algebraist" is a much vaguer term than "analyst" or "topologist". A category theorist could be called an algebraist, someone doing finite group theory will be using very different methods than someone doing infinite group theory, and they work with completely different structures than someone doing ring theory or galois theory. So the question becomes: what about a pursuit makes it "algebraic"? I would say the focus is on some notion of transformation. An action is "algebraic" if it involves pushing some object through a transformation to see what happens. An algebraist studies the way these transformations interact with each other. Turning it around "algebraic [insert mathematical field here]" is the study of a given class of objects (those of the mathemtical field we are "algebraizing") by studying how the objects move under these transformations. So, I would say an algebraist studies transformations. This is my principal reason for calling category theorist algebraists: when it comes down to it, they aren't studying categories, they are really studying functors and natural transformations ways that categories can be transformed. Also, you say Quote:
Cheers, Cory 
Re: Abstract Algebra I failed to respond to the following statement in the above, and I'm feeling rather philosophical (and not particularly sleepy... and also, apparently, verbose) today, so I'll say something about this. Quote:
In fact, whenever you have a transitive, reflexive relation, you have morphisms, and vice versa. The idea of morphism has very much permeated all of math. Even if it's not (explicitly) being used in an algebraic sense, this categorical language is becoming more and more common, because it very nicely captures something all of matehmaticians do: apply a certain type of function to our objects. What type of function? One that preserves the "interesting" structures of our object. I find it hard to believe that such a general and pliable notion is useless for any mathematician. *There are some really great results that prove the colorability of whole classes of graphs, simply by making use of composition of morphisms, and apparently graph homomorphisms are being used to precisely and neatly say things that could only be said using rather messy and approximate arguments before. 
Re: Abstract Algebra fair enough, I wish you had taught me algebra. 
Re: Abstract Algebra There's so much more to algebra than groups, rings and fields! In broad terms an algebra is just a pair , where is a set and is a set of operations of finite arity on , in which a number of additional rules may hold governing the actions of the operations. The additional structure that can be placed on a general algebra, such as demanding that certain identities hold in the application of sequences of operators (e.g. associativity etc.) make the concept of an algebra very flexible in what it can be used to model. Along with the familiar objects mentioned above algebras have applications in order theory (lattices), logic (boolean algebras with operators, cylindric algebras etc.), theoretical computer scientists can even use algebra to describe the way computer programs work (Kleene algebras), and there is plenty more besides these examples. With regards the terminology, on an undergrad course it can seem like its just there for its own sake. You prove a lot of stuff that seems like busywork. But this is just because even relatively advanced undergrad/beginning grad courses are really only introductions. They're trying to give you an overview of the tools that are available but they rarely have time to motivate them by going into the problems from which the definitions emerged. 
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(If only I spent that time doing it...) Thanks for your post, pseudonym, that's a really important point; it also explains why "algebraic combinatorics" focuses so much on lattice theory. (At least, if my description of "algebaric ___" is correct in general.) Also, Quote:
I think this shows up in topology a lot; the definition is signficiantly more abstract than anything most students have seen in analysis at that point, and some understanding seems to get lost along the way. The number of questions on math overflow revolving around "Why is topology definted this way" is some interesting evidence for this. 
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Re: Abstract Algebra You all sound very intimidating. I am just about to leave my world of problem sets for this wider, terrifying world. I don't know whether reading this is inspirational of scary. 
Re: Abstract Algebra Hmm... that very well could be the problem... And I do know that my instructors seem to have gotten better at communicating motivation over the past couple of years... perhaps I've just gotten better at understanding it. 
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I can promise it is much more requarding and enjoyable once you start trying to break out of problem sets pursuing an idea (even a fruitless one!) is very exciting, and gives you a much deeper understanding of the thing you're working with than any problem set can. Suddenly seeing a connection (such as noticing a surprising structure show up "in the wild") is a wonderful feeling that is very difficult to get across with problem sets. (Although I certainly have had this happen while working on a problem set.) Of course, problem sets will always be important I never expect to understand a book until I work the problems, and never expect to understand a lecture or paper without working out the proofs on my own, even when they are "trivial" so you'll be able to comfortably hide inside a cozy problem set for a bit whenever you get too afraid of the wilderness. :D 
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