Talk:Abstract algebra

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"This grants the mathematician who has learned algebra a deep sight, and empowers him broadly." This sentence seems a little odd. I suppose that it's meant to convey the advantages of studying algebra, but what exactly is "deep sight", and how can you be broadly empowered? Can it be clarified, or should we remove it? Hermajesty 18:52, 14 January 2006 (UTC)


Is it common usage to call a group "an abstract algebra"? I think of abstract algebra as a field of mathematics which studies algebraic structures such as groups. And the term "abstract" is only used if there is a need to distinguish it from elementary or college algebra. Maybe there's also a confusion with universal algebra? --AxelBoldt

Yes, the term abstract algebra refers to the field of mathematics that studies abstract algebraic structures, not to the structures themselves. Abstract here refers to the fact that you are interested in all the objects that satisfy certain axioms instead of one particular object. For example this is the difference between studying vector spaces, and studying R. This has nothing to do with the difficulty level of the enterprise.
Abstract algebra has many subfields. For example group theory or lattice theory, which study one particular type of abstract algebraic structures. Universal algebra is another subfield of abstract algebra that develop properties or theorems that apply to all abstract algebraic structures. For example in the study of group theory you are lead to the isomorphism theorems. In ring theory your are lead to similar theorems. The idea of universal algebra is then to find a more general formulation that would apply to all abstract algebraic structures.
Category theory can be understood as a generalization of universal algebra. It studies results that apply to all categories, i.e. not only abstract algebraic structures but other things like topological spaces for example.
Hope this helps. Ceroklis 16:12, 29 September 2007 (UTC)

I thought it was a bit odd too. I would have used the term "algebraic system". In universal algebra they would just be called "algebras" (except for modules and vector spaces, which don't qualify because of the external multiplication). I'm not sure what to do about it at the moment. In any case, we need an article on universal algebra. --Zundark, 2001-09-04

From the (principal so far) author of the article on universal algebra: Modules and vector spaces are indeed covered under universal algebra if you fix the ring R that the modules are over. But scalar multiplication is not a binary operation of course; instead, for each element r of R, you have a unary operation "scalar multiplication by r". There's no rule that says, for example, that you must have only finitely many operations!
Somebody should probably explain this in the article on universal algebra, or maybe on modules, but I'm not sure how to organise it now. --Toby Bartels, 2002/04/03
I think it fits in universal algebra, as an additional example. And all the structures that are covered by universal algebra should have a link to universal algebra. Can you also deal with topological groups in universal algebra? If not, then that should be mentioned as well. AxelBoldt
Well, I'll put it in there, but I don't know when I'll get to it -- I'm having lots of fun looking around here.
As for topological spaces, universal algebra doesn't handle them as such. However, if you start with topological spaces as given, then you can define universal algebra in the category of topological spaces analogously to defining universal algebra in the category of sets, just as you define a topological group (that is a group in the category of topological spaces) analogously to defining a group (that is a group in the category of sets). Every time that the definition of an algebraic system calls for a set, you replace that with a topological space; and every time that the definition calls for a function, you replace that with a continuous map. So topological groups aren't covered by universal algebra any more than they're covered by group theory, but they are covered by topological universal algebra, just as they are (obviously) covered by topological group theory.
So I suppose that I should add a comment to that effect.
-- Toby Bartels

Contents

[edit] additionalitems studied in abstract algebra

You should add ring modules,vector spaces,and algebras at least.

   Remag12@yahoo.com 05:42, 10 January 2006 (UTC) S. A. G.

[edit] Abstract algebra removed from Wikipedia:Good articles

Abstract algebra (edit|talk|history|links|watch|logs) was formerly listed as a good article, but was removed from the listing because the article lists none of its references or sources --Allen3 talk 20:36, 18 February 2006 (UTC)

[edit] Not formally a good article yet

I took this up to review it just as it got coincidentally removed from the list. There should've been a Template:GAnominee placed at the top of this page when it was nominated, too. Oh well, here's my comments anyway.

This is, essentially, an important article, that should bring a clear and engaging overview of a field of mathematical activity. Many readers of the Mathematics article will, as their next port of call, fancy a dip into some of the various branches, and the Abstract Algebra article will sometimes be first stop - readers will be enticed by the image of a Rubik's Cube or the mention of root2.

I think that the article currently pitches to undergraduate mathematicians and higher, and that it remains quite opaque for readers with less experience than that. It specialises far too soon and makes few concessions to the more vivid elements of its subject. As it resides "near the front of" the mathematics Wikirealm, its duty is to offset its specialism with a much gentler pace, more informal language and vividity in the exposition. That said, the existing prose is very eloquent and much of it definitely deserves to remain.

The "Example" has a superb introductory sentence more suited to the whole article's introduction. The actual example assumes a lot of knowledge. I think that a more appropriate level of example would be (for instance) one in which several disparate objects are shown to have inverses relative to an identity. The notions of homomorphism and isomorphism so crucial to abstract algebra should also be expanded upon, preferably with an example.

Much more should be made of abstract algebra's branches. Prominent or thriving subdivisions such as group theory or boolean algebras should be expanded upon in an elementary and vivid way, most easily with examples. Or at the least, there should be pointers to tangible articles such as examples of groups. As it is, the history is generalised down into a single sentence, and the examples form a list of algebraic structures. Material currently in the introduction needs breaking off into sections, and I suspect some of it can go into "history", with a little expansion.

Examples of abstract algebra's usefulness need expanding upon.

Although vector spaces are listed, neither Linear algebra nor its relationship to abstract algebra are mentioned. The distinction of representation theory is not clear: what is concrete about it that distinguishes it from abstract algebra?


The references and external links are excellent.

I could not have passed the article. It is stable, factually accurate and neutral, but it is not yet sufficiently broad or comprehensible. Topology and Calculus are currently useful comparisons for this article. Please feel free to call upon me for my comments prior to resubmission. --Vinoir 04:03, 27 April 2006 (UTC)

By the way, the guidelines for good articles are here. --Vinoir 04:19, 27 April 2006 (UTC)
I'd god further than that. I don't think it even reaches B-Class. --Salix alba (talk) 11:01, 16 June 2006 (UTC)


Shouldn't this article mention some of the mathematicians that had a hand in forming modern algebra? (Galois, Hilbert, Noether, et al.) shotwell 14:04, 22 September 2006 (UTC)

[edit] What questions should this page answer?

In the interest of improving this page, here are some proposed questions that I could imagine a reader of the article either coming here to find the answer, or being pleased to discover the information. Feel free to add questions. -- Jake 21:18, 5 October 2006 (UTC)

  • When was the term "Abstract algebra" coined?
  • What was the order of the historic development of the various algebraic systems (at this level of abstraction)?
  • What value does has the notion of "Abstract algebra" contributed to mathematics?
  • Do we have courses on Abstract algebra, because of the topic, or do we have the notion because the courses needed a name?
  • Who were / are the big names in the field and what were their contributions (at a high level, not to duplicate content in the other articles)?
  • What distinguishes abstract algebra from related fields?
  • Are there people who would consider themselves (abstract) algebraists? Or would an individual be more likely to describe themselves as, say, a ring theorist?
  • Beyond a simple listing of the subfields, what can we way about how they are qualitatively or quantitatively different?
  • ...

[edit] Commented text

A recent edit commented out some text, unsure where it should go (if anywhere). Here it is.

Formal definitions of certain algebraic structures began to emerge in the 19th century. Abstract algebra emerged around the start of the 20th century, under the name modern algebra. Its study was part of the drive for more intellectual rigor in mathematics. Initially, the assumptions in classical algebra, on which the whole of mathematics (and major parts of the natural sciences) depend, took the form of axiomatic systems. Hence such things as group theory and ring theory took their places in pure mathematics.

Examples of algebraic structures with a single binary operation are:

More complicated examples include:

See algebraic structures for these and other examples.

I'm not an expert, but it seems that these links should still be present, if only at the end of the article. Geometry guy 02:24, 28 March 2007 (UTC)