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I've listed this article for peer review to prepare it for a featured article candidacy. I was hoping to get feedback on its current status and what improvements are required to fulfill the featured article criteria.

Thanks, Phlsph7 ( talk) 17:05, 14 June 2024 (UTC) reply

• @ Phlsph7: This has been posted for over a month without comment. Are you still looking for comments, or can this be closed and nominated to FAC? Z1720 ( talk) 15:48, 24 July 2024 (UTC) reply

  @ Z1720: Thanks for the reminder! I now posted to the WikiProject Mathematics to attract more reviewers. I was hoping to keep it open a little longer to see if someone responds. Phlsph7 ( talk) 07:44, 25 July 2024 (UTC) reply

  @ Phlsph7, @ Z1720 Maybe we can start from the sources, images, and other FAC criterias to be checked before heading to the FAC? Dedhert.Jr ( talk) 09:39, 25 July 2024 (UTC) reply

  Hello Dedhert.Jr, that sounds good, thanks for taking a look! Phlsph7 ( talk) 10:37, 25 July 2024 (UTC) reply

  To be honest, I'm not actually an expert in FAC, and I don't even have one. You can try to find someone interested in this topic, or ask non-mathematics users to review your article, from which they may find something that will be added or should be removed. Dedhert.Jr ( talk) 10:42, 25 July 2024 (UTC) reply

  If you or someone else would be willing to read through the article and provide some feedback, this would be helpful to consider potential problems that I may have overlooked. The peer review can be done casually without precisely following the FA criteria and your background in mathematics would be particularly valuable in identifying possible improvements. Phlsph7 ( talk) 11:34, 25 July 2024 (UTC) reply

• My first take is that the second image in {{ multiple image}} is not appropriate; ${\displaystyle \mathbb {C} ,\ \mathbb {N} ,\ \mathbb {Q} \ {\text{and}}\ \mathbb {Z} }$ are covered in elementary algebra classes. Perhaps ${\displaystyle O(n)}$ would be a better choice? — Preceding unsigned comment added by Chatul ( talk • contribs)

  Thanks for the input! I assume you are referring to the Orthogonal group ${\displaystyle O(n)}$. This could be done but I think it is not the easiest way to introduce abstract algebra. Most introductions to abstract algebra start with more familiar examples of algebraic structures, like the ring of integers. Its true that there is an overlap here with elementary algebra. The main difference is that elementary algebra studies how equations formulated within these algebraic structures can be solved while abstract algebra studies and compares the algebraic structures themselves. Phlsph7 ( talk) 16:16, 25 July 2024 (UTC) reply

  Yes, Orthogonal group. How about a permutation group? -- Shmuel (Seymour J.) Metz Username:Chatul ( talk) 17:07, 25 July 2024 (UTC) reply

  How about using the Rubik's Cube group as an example of abstract algebra?

  

  Elementary algebra studies which values solve equations formed using arithmetical operations, especially polynomial equations.

  

  Abstract algebra studies algebraic structures, like the Rubik's Cube group, a group ${\displaystyle (G,\cdot )}$ that represents the structure of the Rubik's Cube mechanical puzzle. Each element of the set ${\displaystyle G}$ corresponds to a cube move, which is the effect of any sequence of rotations of the cube's faces.

  -- Shmuel (Seymour J.) Metz Username:Chatul ( talk) 13:51, 26 July 2024 (UTC) reply

  Permutation groups could work if the article was written for mathematicians. But since it is written for a general audience, we would probably have to explain to them first what a permutation is before they can understand that a permutation group is an example of an algebraic structure.

  How about SO(2) as the second image? Perhaps a circle with a curved arrow. I believed that it is simple enough for a layman to understand and that it is not normally covered in elementary algebra. -- Shmuel (Seymour J.) Metz Username:Chatul ( talk) 10:24, 28 July 2024 (UTC) reply

  It's very difficult to find a representative image for abstract algebra that gives the average reader visual confirmation that they've arrived at the right page. Ideally, the image should be self-explanatory without the caption. I fear that, by using a concrete application, like an image of a Rubik's cube or a circle with arrows in it, we give the reader a false impression even if we correctly explain it in the caption. Apologies for dismissing your different lead image ideas. I agree with you that the current abstract algebra image is not ideal either. Maybe the best option would be to remove it and only have the elementary algebra image. Phlsph7 ( talk) 15:17, 28 July 2024 (UTC) reply

  I like the idea of using Rubik's cube as an example. It's not particularly representative of Algebra in general (see MOS:LEADIMAGE), so I think it would be better to include it somewhere else in the article, maybe a short paragraph in the section "Applications". I'll look into it. Phlsph7 ( talk) 16:08, 26 July 2024 (UTC) reply

  I found a way to mention Rubik's cube together with the image in the section "Applications". Phlsph7 ( talk) 07:44, 27 July 2024 (UTC) reply

This article is well-written in a clear style and it is masterfully balanced. I especially appreciate the large number of pictures and the fact that it doesn't focus on only one meaning of the word "algebra". However, there are some issues.

(I) When I read the descriptions of elementary algebra in this article, as a reader, I am left with the impression that elementary algebra is about solving single polynomial equations. Two examples:

1. Elementary algebra is interested in polynomial equations and seeks to discover which values solve them.
2. Arithmetic is the study of numerical operations and investigates how numbers are combined and transformed using arithmetic operations like addition, subtraction, multiplication, and division. For example, the operation of addition combines two numbers, called the addends, into a third number, called the sum, as in 2 + 5 = 7. Elementary algebra relies on the same operations while allowing variables in addition to regular numbers.
3. There is no mention of exponentiation, roots, or logs in the context of elementary algebra in this article.

I think we should mention that exponentiation, roots, logs, and simultaneous equations can come up in problems under the scope of elementary algebra.

Thanks for pointing this out. I adjusted the two sentence to make them more general while also mentioning exponentiation, roots, and logarithms. We have to find a middle way here since they should be mentioned but at the same time shouldn't be at the center of the discussion. Phlsph7 ( talk) 11:12, 26 July 2024 (UTC) reply

(II) Change this excerpt:

...it is possible to express a general law that applies to any possible combinations of numbers, like the principle of commutativity expressed in the equation...

To this:

...it is possible to express a general law that applies to any possible combinations of numbers, like the commutative property of multiplication, which is expressed in the equation...

Or something similar.

Changed. Phlsph7 ( talk) 11:15, 26 July 2024 (UTC) reply

(III) I am a bit worried about this sentence in the lead, which seems worded strangely:

They relied on verbal descriptions of problems and solutions until the 16th and 17th centuries, when a rigorous mathematical formalism was developed.

After looking into this, I think "verbal descriptions" just means that they wrote down equations using words and abbreviations instead of symbols. I think this is not how most people would interpret this sentence.

I reformulated the sentence, I hope it is clearer now. Phlsph7 ( talk) 16:17, 26 July 2024 (UTC) reply

I like your edit, but I added the word "symbolic" to it and now I like it even better. Mathwriter2718 ( talk) 16:29, 26 July 2024 (UTC) reply

I agree, the word helps sharpen the contrast. Phlsph7 ( talk) 16:43, 26 July 2024 (UTC) reply

(IV) Maybe the section on universal algebra can be written better?

1. Surely at least one of these two statements is true: (a) quasi-identities can be explained better or (b) they don't belong in this article.

   Unfortunately, I'm not sure how to explain it better. I moved the paragraph to a footnote. This way, it's still there for the curious but is less likely to confuse the average reader. Phlsph7 ( talk) 16:34, 27 July 2024 (UTC) reply

   Seems like the best solution at this time to me. Mathwriter2718 ( talk) 17:55, 27 July 2024 (UTC) reply

2. The description of subalgebras reads backwards in the sense that instead of telling us what a subalgebra is, it just apparently lists a few facts about them:

Another tool of comparison is the relation between an algebraic structure and its subalgebra. If ${\displaystyle \langle A,\circ \rangle }$ is a subalgebra of ${\displaystyle \langle B,\circ \rangle }$ then the set ${\displaystyle A}$ is a subset of ${\displaystyle B}$. A subalgebra has to use the same operations as the algebraic structure and they have to follow the same axioms.

I reformulated the passage to have the explanation in a more natural order. Phlsph7 ( talk) 16:34, 27 July 2024 (UTC) reply

(V) There are a lot of encyclopedia articles in the references and notes, which would seem to violate WP:TERTIARY. I wonder if this came up in the GA reviews?

Best of luck! Mathwriter2718 ( talk) 19:35, 25 July 2024 (UTC) reply

One GA reviewer brought up the encyclopedia articles in the references and notes at Talk:Algebra/GA1#Referencing_and_other, but they didn't seem to mind. To be specific, the policy that worries me is:

Articles may make an analytic, evaluative, interpretive, or synthetic claim only if it has been published by a reliable secondary source.

Which is actually listed under WP:SECONDARY. It looks to me like most of the "analytic, evaluative, interpretive, or synthetic claim[s]" made in this article have citations to a non-encyclopedic source even if there is also an encyclopedic source, but there are a few exceptions, for example this sentence:

Mathematicians soon realized the relevance of group theory to other fields and applied it to disciplines like geometry and number theory.

It shouldn't be super bad to fill in the gaps. Mathwriter2718 ( talk) 20:57, 25 July 2024 (UTC) reply

I added a secondary source for this claim in the history section. For articles on narrow topics, I would agree with you that it is often preferable to minimize the use of general encyclopedias because of the importance of getting specific details right. But I feel that the situation is different for articles on very general topics like this one, especially when there are good technical encyclopedias are available. For wide-scope articles, it's not so much about particular in-depth details but about getting the overview right. A while back, there was a similar discussion about the FA nomination of the article Logic, see Wikipedia_talk:Featured_article_candidates/archive90#Usage_of_tertiary_sources_in_the_article_Logic and Wikipedia:Peer_review/Logic/archive3.

Thanks for all the improvement ideas! Phlsph7 ( talk) 07:31, 28 July 2024 (UTC) reply

Thanks for all your work improving this article, and good luck again on getting this to FA status! Mathwriter2718 ( talk) 11:45, 28 July 2024 (UTC) reply