Talk:Abstract algebra

Simplifications vs Rigorous Correctness

While to mathematicians, Abstract Algebra is in the tradition of Hilbert's pursuit of rigor, it can quickly become "opaque" to the layman. So how to bridge the snippy criticism of mathematicians about X being left out, etc. with the bewilderment of laymen if X were included? Is the Wiki article to be an undergraduate text book for mathematics majors or a general reader's guide to what the general reader can legitimately view as difficult abstract topic? God bless any writer attempting to thread that needle. — Preceding unsigned comment added by 98.169.63.134 (talk) 02:45, 14 January 2010 (UTC)

Application of abstract algebra?

It seems abstract algebra only has application in mathematics to develop more complex concept such as algebraic number theory or algebraic topology.kongshengxin (talk) 06:41, 30 June 2012 (UTC)

More applications should be mentioned, particularly to computer science in areas such as error correction codes and cryptography. --136.160.159.161 (talk) 17:30, 3 August 2016 (UTC)

Clean of the flowchart

Am I the only one to find the flowchart unreadable? For me, there is no box, and arrows everywhere, making very hard to figure which text refers to which arrows. Plus, arrows are in all directions. I guess it would be better to have shorter text, draw boxes, start arrow from boxes edges and try to arrange this stuff so that all arrows are from down to up with no exception. Sedrikov (talk) —Preceding undated comment added 10:25, 11 July 2012 (UTC)

Removal of diagram

I pulled the diagram "File:algebraic_structures.png". It really represents one person's conception of these relationships, and is not really encyclopedic material. I appreciate the poster's work and intentions, but I think the diagram is not as helpful as intended. This is why I have elected to pull it from the article. Rschwieb (talk) 13:13, 10 August 2012 (UTC)

Examples of monoids? Really?

Functions under composition and matrices under multiplication are only monoids if the domain and codomain are equal and if the matrices are square (and the size is fixed), respectively. Otherwise, they are just categories. We must strive to not be misleading. --Eduardo León (talk) 23:26, 12 January 2013 (UTC)

Beyond that, the paragraph failed to mention the functions had to be linear endomorphisms of a finite dimensional vector space. The first two paragraphs were misbegotten in the first place, and they didn't really exemplify anything, so I took them out. Rschwieb (talk) 13:33, 13 January 2013 (UTC)

Motivation of a revert

I have reverted a recent edit for reasons that are too long for an edit summary:

• The edit replaced "abstract algebra is a usual name for the subarea ..." by "abstract algebra is the subarea ...". This would be fine if "abstract algebra" would be a well established subarea. But this is not the case. For example the Mathematics Subject Classification does not mention "abstract algebra" at all. It is important to warn the reader in the lead that the term "abstract algebra" is somehow controversial by itself.
• The edit removed "for themselves" in "that studies for themselves algebraic structures ...". Again, it is highly controversial to suggest that the study of an algebraic structure in view of applications is "abstract algebra". For example, the study of Rubik's cube belongs to group theory, but it is a wrong idea to consider it as "abstract algebra".

D.Lazard (talk) 15:18, 27 April 2013 (UTC)

If the consensus is that abstract algebra is not a well established subarea, then that should be stated directly rather than being hinted at. The following "History" section is where that already happens, so it's not like I'm being revisionary. And what is added by the parenthetical remark "(occasionally called modern algebra)" when modern algebra clearly redirects here and the evolution of both terms are spelled out in the following section? If nobody else objects, I would like to remove this issue from the lede and include the other grammatical changes that I made. Cutelyaware (talk) 21:56, 16 August 2019 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Abstract algebra/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

History needs expansion, plus more on motivation/applications. Tompw 13:04, 21 November 2006 (UTC)

Last edited at 13:04, 21 November 2006 (UTC). Substituted at 01:43, 5 May 2016 (UTC)

Role of the article

Lots of nice stuff happening in the history section for the 30 kb vital articles drive. But I think the article is missing something to bring it to life/give it purpose. I think this could be a nice place to summarize the different subfields of abstract algebra, and give attention to those fields that are of interest in undergraduate (and higher) math.

For example, the article doesn't get across how groups are pretty much one of the most important concepts in modern mathematics, encountered in every year of an undergraduate math degree, while a magma might not even be defined.

I'd think there should certainly be sections on group theory, ring theory, field theory, linear algebra, module theory and algebras. In each section there should be examples of each type of algebraic structure, and fundamental/important results.

Of course it will not (and cannot) be comprehensive, but as it stands I don't see why there should be a summary article for all of abstract algebra with so little detail.

Zephyr the west wind (talk) 11:09, 29 August 2022 (UTC)

Well, the issue is that "abstract algebra" is a non-field. You can see this in the discussion of math SE tags; as T.. says, "abstract algebra" as a label is redundant and can in all cases be replaced by a more specific term such as group theory, commutative algebra, representation theory, or category theory. The only real use of the term nowadays is as the title of undergraduate university courses. There are some citations for "Journal of Abstract Algebra" but all those papers seem to be random nonsense from SCIgen.

As far as discussing the actual structures, I think the individual Wikipedia pages do a much better job and the discussion of each structure in this article should be limited to a paragraph. Defining a group is alright, but I don't think we need to list examples of groups or important results - those are all in Group (mathematics). The notable examples and theorems are also listed on List of abstract algebra topics.

I've been working on the history to explain where this term comes from and why the undergraduate courses include the topics they do. Kleiner describes his primary audience as "teachers of courses of abstract algebra", although he aimed to make the material accessible to high school math teachers and advanced undergraduates, and I think that should be the audience of this article as well. I think what would improve the article further is a discussion of teaching methods. There are various articles, [1] [2] [3] [4] [5]. Also this Chinese article on "Ideological and Political Teaching in Abstract Algebra" (quote: "For example, China and Kazakhstan are two independent countries, like the addition and multiplication of rings"), and this article on acceptance of abstract algebra in the USSR. Mathnerd314159 (talk) 16:25, 29 August 2022 (UTC)

Sure, I'm happy with this response. I realized the article would end up bloated if I did the same for each structure. The real difficulty of this article is not really in finding content for it, but in defining its scope.

I do think yours is a nice direction to take the article, but I don't think this would make the article a vital article - and the right course of action might be to demote this article from being vital.

That is a hilarious quote. Zephyr the west wind (talk) 18:15, 29 August 2022 (UTC)

Alright, proposed a swap. I'm tired of reading through Kleiner and the article is past 30kb so I'll call my work done for purposes of the drive. Mathnerd314159 (talk) 21:22, 29 August 2022 (UTC)