Talk:Abstract algebra

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I've just fleshed out the chapter on ideals, hopefully this isn't too controversial. I agree with earlier comments that the structure is lacking - e.g. no connections between modules and vector spaces. There's also the question of who this is aimed at - advanced high school, undergraduate, graduate / reference and whether accomodating these is easier if using a wiki rather than just a book. The other question is also to what depth one wants to study each subject - too much group theory might be annoying to someone only interested in commutative algebra. Presenting everything in the utmost generality for the sake of non-commutative algebra would obscure the simple case, but not enough would naturally annoy the non-commutative algebraist. The first example isn't too problematic - the reader can simply omit the complicated group theory, and having a massive wikibook doesnt have the same problems as having a massive dead-tree book - but the second is a bit harder because it influences the presentation in an essential way. A similar question might be whether to use category theory, or add in a geometric perspective (ala Atiyah-McDonald and Eisenbud), again these might affect the presentation quite fundamentally. My answers to these questions are probably quite biased - would be good to hear other perspectives --Drufino 22:59, 29 April 2007 (UTC)

This Abstract algebra book seems to be more like a collection of related topics rather than a planned literature. It assumes too much prior knowledge and all the modules are still quite immature. I'd like see a little more structure to it, maybe it should have been called a first course in Abstract algebra and it should assume knowledge of basic linear algebra? Xiaodai 00:56, 22 Aug 2004 (UTC)

Of course the modules are still immature, nearly everything here needs work (not just in this section)

Some sections do assume some prior knowledge, and some do not. The section on Groups for example should be able to be picked up with just a relatively small amount of prior mathematical knowledge. Some of the stuff is quite "disgusting" and needs a lot of work.

The linear algebra stuff needs merely to be bumped up the top, as it shouldn't be too difficult to understand.

Why do you want it to be called "a first course in abstract algebra"? Dysprosia 02:05, 22 Aug 2004 (UTC)

It's very hard trying to cover everything in Abstract algebra all at once, we might want to break the book into smaller books. It's very dangerous calling the book just Abstract Algebra, it's better to do the elementary stuff in a "first course" book then more advaced stuff in a "second course" book and so. Xiaodai 08:01, 24 Aug 2004 (UTC)

Hello I'm new to Wikibooks (Been on wikipedia for a while). I have a few humble sugestions:

• Much of this material needs to be made more accesible. This sort of text ought to develop the ideas more gradually, with more examples and more explanation.
• Category theory really should not be in here. It is way too advanced really and might put people off.
• We should have some sort of dependencies table or at least each chapter should say what knowledge it presumes.
• We should have exercises at the end of each chapter and some sort of answers section (preferably it should have a slowdown built in to encourage people to work things out for themselves).
• We should include a section on naive set theory together with a cautionary exhibition of Russels paradox.

I'm thinking of starting a wikibook on Galois theory and maybe one on Set theory so I may be around some more. Barnaby dawson 21:26, 25 Aug 2004 (UTC)

1. Barnaby, the books still need a lot of work. I just said that above. They will get better with time and effort.
2. It should be included in a book on abstract algebra. The stuff isn't that hard. If we stratify the abstract algebra outline more, it will have a place
3. Ok
4. see 1.
5. see Discrete mathematics/Set theory, though it should be here too since it overlaps, I think I need to go rip out the DM book a bit more.

Cool, a section on Galois theory would be very welcome. When I get a chance to, I'll go and radically restructure everything, or something. Dysprosia 22:16, 25 Aug 2004 (UTC)

--- Outline of Basic Ring Theory?

In the ring theory section, one thing that would help unify the concepts that are being presented would be to spend some time developing the concept of the Division Algorithm, GCDs, and the Euclidean algorithm. It appears that this is begun under the "Modular Arithmetic" module of the discrete math book. However, the strong connection between relative primality/irreducibility and existence of multiplicative inverses is not really described.

Having done this, it becomes possible to generalize the Division Algorithm to polynomials of a single variable. This allows for the Euclidean algorithm to be carried over and we can develop the idea of quotient rings. (And of F[x] as a field.) This leads naturally into the development of ideals.

Finally, coverage of ring homomorphisms and the Isomorphism Theorems needs to be introduced. This would be a good start on getting to more advanced topics like modules, lattices, and eventually, polynomials in multiple variables.

All in good time, I know, but it might be wise to spend a moment thinking about what the overall organization of ring theory is like. I personally see the initial definitions of rings, fields, and integral domains as being the "easy" part and the development of the Division Algorithm, GCDs, the Euclidean Algorithm, followed by polynomial rings, ideals and quotient rings, and ring homomorphisms as the "meat" of basic ring theory.

Thoughts? --Ashsong 06:44, 21 Dec 2004 (UTC)

Well, be bold, and add the stuff! That's what the wiki's all about. Though, I've removed the division algorithm, etc., from having their own modules (they can and should be dealt with in their respective modules when necessary) and cleaned up the structure a bit. Dysprosia 07:37, 21 Dec 2004 (UTC)

Do you have any favorite wikibooks that I ought to look at in order to get a better idea of the style that you'd like to see? --Ashsong 15:27, 21 Dec 2004 (UTC)

[edit] Proposed name change

I'd like to change all the subpages of this book so that they begin with "Abstract Algebra/" rather than "Abstract Algebra:" to make them consistent with the new naming convention. I trust this will be non-controversial, but note it here to allow for comments before the move, Jguk 07:01, 7 April 2006 (UTC)

hey can some1 tell me what "Ris are predicate symbols of some arity and ν is an interpretation of these symbols on the set." means?Vandal 0nly account 07:52, 22 August 2007 (UTC)

[edit] Number theory is not prerequisite

Requiring that one knows number theory before beginning is a little silly to me. Algebra is far more fundamental to everything, while number theory is a bunch of cool results that aren't useful, many times.