Talk:Abstract algebra/Archive 1

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You should add ring modules,vector spaces,and algebras at least. [email protected] 05:42, 10 January 2006 (UTC)

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)

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

Formal definitions of certain

Examples of algebraic structures with a single binary operation are:

• magmas,
• quasigroups,
• monoids, semigroups and, most important, groups.

More complicated examples include:

• rings and fields
• modules and vector spaces
• algebras over fields
• associative algebras and Lie algebras
• lattices and Boolean algebras

See

I'm not an expert, but it seems that these links should still be present, if only at the end of the article.

"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 — Preceding unsigned comment added by AxelBoldt (talk • contribs) 16:46, 4 September 2001 (UTC)

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 — Preceding unsigned comment added by Conversion script (talk • contribs) 14:51, 25 February 2002 (UTC)

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

talk • contribs

) 18:45, 3 April 2002 (UTC)

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 — Preceding unsigned comment added by AxelBoldt (talk • contribs) 08:20, 4 April 2002 (UTC)

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.

--

talk • contribs

) 02:24, 8 April 2002 (UTC)

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?

• Do the meanings of
  Set Theory ? A quote from Pierre Cartier, "Bourbaki got away with talking about categories without really talking about them. If they were to redo the treatise [Bourbaki's not Whitehead's], they would have to start with category theory. But there are still unresolved problems about reconciling category theory and set theory
  ."

• ... --Firefly322 (talk) 09:52, 11 March 2008 (UTC)

Theres something I can do but this is not one of them. Here are some examples: 15 meters to millimeters?, 3.5 tons to pounds?, 6800 seconds to hours? could anyone help me understand how th you get the answer for these? —Preceding unsigned comment added by 72.94.239.161 (talk) 00:45, 15 September 2008 (UTC)

The correct place to ask this type of question (since it is not about the contents of this article) is the Mathematics Reference Desk. Go to that page, read the instructions, then post your question there. Gandalf61 (talk) 08:59, 15 September 2008 (UTC)

If I state "X is derived from Y", and quit then I am attempting to state a fact, and it deserves some citation if it is not general knowledge. If I state "X is derived from Y, and here is the evidence for it" then I have a thesis statement. If so, the citations are needed for the evidence I present ostensibly supporting my thesis. So the first [citation needed] is perhaps misplaced, since the ensuing portions of the article go to great lengths to support the thesis, unfortunately in "Early Group Theory" the citations needed are lacking. — Preceding unsigned comment added by 98.169.63.134 (talk) 02:45, 14 January 2010 (UTC)