Wikipedia:Featured article candidates/Algebra/archive1

The following is an archived discussion of a featured article nomination. Please do not modify it. Subsequent comments should be made on the article's talk page or in Wikipedia talk:Featured article candidates. No further edits should be made to this page.

The article was promoted by Gog the Mild via FACBot (talk) 28 October 2024 [1].

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Algebra

Nominator(s): Phlsph7 (talk) 08:45, 6 August 2024 (UTC)

Most people are familiar with algebra from their school days, where they learned to solve equations like $x^{2}-3x-10=0$. However, there is also a more abstract form of algebra, which is of particular interest to mathematicians because it provides a general framework for understanding operations on mathematical objects. Thanks to Bilorv for their in-depth GA review and to Mathwriter2718 and Chatul for their peer reviews. Phlsph7 (talk) 08:45, 6 August 2024 (UTC)

RoySmith (Support)

• As an administrative note, I noticed this page had been semi-protected since 2008. 16 years of protection seemed excessive, so I unprotected it.
• I'm still looking at this, but I do want to say that it's a joy to see a math article which is so approachable. My training is in engineering; I'm a user of math, but not a mathematician. Most math articles (Lie algebra being a good example) make my eyes glaze ever before I get past the first sentence. In this article, I'm down to Linear algebra and I'm still following every detail. This is wonderful!

  Hello RoySmith and thanks for reviewing the article! The parts on abstract and universal algebra will get a little more challenging but this is not entirely avoidable and I hope they are still accessible enough to grasp the main ideas without feeling overwhelmed. Phlsph7 (talk) 15:08, 6 August 2024 (UTC)

  Yeah, I'm working on Abstract algebra now. Slower going than before, but I'm still hanging in there :-) RoySmith (talk) 15:11, 6 August 2024 (UTC)

• The word algebra comes from the Arabic term الجبر (al-jabr), which originally referred to the surgical treatment of bonesetting You can't just leave the reader hanging without giving at least some explanation of how we got from bonesetting to a high-school math class. You link to Traditional bone-setting, but that doesn't say anything about it. I did a bit of searching. "The Origin of the Term "Algebra" on JSTOR". jstor.org. Retrieved 6 August 2024. talks about this a bit while "Simplifying equations in Arabic algebra". sciencedirect.com. Retrieved 6 August 2024. suggests the connection may be entirely accidental. Either way, I think it's worth a sentence or two.

  That's a good idea and the sources are helpful. I put it in a footnote since there is no consensus on the exact meaning. Phlsph7 (talk) 16:01, 6 August 2024 (UTC)

• x-y-pair do you need both hyphens? I would think "x-y pair".

  Done. Phlsph7 (talk) 16:07, 6 August 2024 (UTC)

• An equation is linear if ... no operations like exponentiation, extraction of roots, and logarithm are applied to variables why the equivocation, i.e. "operations like"? Which operations are like those and which are unlike? My understanding is that an equation is linear if there's no power greater than 1, and things like logs and roots get included in that implicitly via their Taylor series. I think this would be better written as an explicit list of operations that are allowed, rather than a vague "stuff like this isn't allowed".

  Done: I added the general form and cut the "stuff like this isn't allowed" down to a short side remark. Phlsph7 (talk) 16:27, 6 August 2024 (UTC)

• An operation[k] is associative if the order of several applications does not matter, i.e., if (a circle b) circle c ... This is the first time you use the circle notation. A little earlier when you introduce N for Natural Numbers, you do a good job of explaining what the notation means; you should do similarly here. My understanding is that it's just "an arbitrary binary operation", but that should be clarified.

  That explanation is given in a footnote, which was unfortunately positioned in a rather unintuitive place. I moved it right after the first use of the circle symbol so that the connection is clearer. Phlsph7 (talk) 08:18, 7 August 2024 (UTC)

• An operation admits inverse elements... explain what it means to "admit" an element.

  I reformulated the expression to make it simpler. Phlsph7 (talk) 08:24, 7 August 2024 (UTC)

• The natural numbers ... contain only positive numbers Why not "positive integers"?

  Changed. Phlsph7 (talk) 08:24, 7 August 2024 (UTC)

• Group theory is the subdiscipline of abstract algebra studying groups. studying -> which studies.

  Reformulated. Phlsph7 (talk) 11:00, 7 August 2024 (UTC)

• I'm not sure how File:Magma to group4.svg relates to the rest of the article, or at least to the text it's near. It's near a section that talks about rings and fields, but the diagram only shows relationships for groups. Perhaps it should go with the following paragraph, where magmas et al are discussed?

  You are right, I moved it to the following paragraph. I expanded the caption to make it easier to see how it is relevant to the discussion. Phlsph7 (talk) 11:00, 7 August 2024 (UTC)

• They differ from each other in regard to the types of objects they describe and the requirements that their operations fulfill To me, this is the key sentence in this whole section. To go back to my comment about my eyes glazing over when I read articles like Lie algebra, this lays out the logical foundation that helps me read " a Lie algebra (pronounced /liː/ LEE) is a vector space ... together with an operation called the Lie bracket, an alternating bilinear map ... that satisfies the Jacobi identity}} and really start to get my head around what it's saying. By analogy, I have a rudimentary knowledge of Spanish, but I can usually read something well enough to say, "OK, that's a conjugated verb. I don't recognize the verb, and I'm not sure about the tense, but at least I can get past that and keep going with the sentence, knowing I can always go back and look up the verb later". The same thing here. I don't know what an "alternating bilinear map" is, but with your explanation in mind, I can say, "OK, I don't know what that is, but at least I recognize it's describing "the requirements the operation fulfills" and I can keep making progress, knowing I can come back later and dig deeper. The point of this rambling note is just to say that I think this sentence needs more prominent placement, perhaps in the first paragraph of this section. Then you can still conclude the section by saying that you've only described the three most basic structures, and lots of other ones exist, such as magmas, etc.

  I found a way to mention this characterization in the first paragraph. I very much agree with you about the accessibility problems of several math articles. Some of them read as if they were written primarily for mathematicians, which becomes a problem specifically for the lead if an educated non-expert reader can't figure out what the topic of the article is. Phlsph7 (talk) 11:22, 7 August 2024 (UTC)

(I'm done with Major Branches. I'll pick up with History another day)

I couldn't stay away, so I finished this up today. I really can't find anything else to complain about in the rest of the article. I'll just leave you with a couple of suggestions which you can take or leave at your pleasure. One is that in Other branches of mathematics where you talk about algebraic solutions to geometric problems, you might want to mention that origami has been used to solve algebraic problems using geometry, see for example https://sites.math.washington.edu/~morrow/336_09/papers/Sheri.pdf. The other is that I don't think you can talk about Gerolamo Cardano without at least mentioning that he has been credited with inventing (or at least accepting the existance of) imaginary numbers.

I added a short side remark about origami and mentioned imaginary numbers. Thanks a lot for all the helpful suggestions! Phlsph7 (talk) 12:16, 7 August 2024 (UTC)

• I'm not qualified to review this for the quality of the research or comprehensiveness, but I'm happy to give my support for general structure and "prose is engaging and of a professional standard".

Image review

• File:Muḥammad_ibn_Mūsā_al-Khwārizmī.png: the fine print on the licensing tag suggests a cropped image like this might not be covered by the copyright exception

  I removed the image since there were also other concerns about it in the comments below. Phlsph7 (talk) 08:39, 7 August 2024 (UTC)

• File:Rhind_Mathematical_Papyrus.jpg needs a US tag

  I added an US tag. Phlsph7 (talk) 15:44, 7 August 2024 (UTC)

• File:Francois_Viete.jpg needs a US tag, and why is this linked to the article? Ditto File:Frans_Hals_-_Portret_van_René_Descartes.jpg. Nikkimaria (talk) 05:05, 7 August 2024 (UTC)

  I added US tags and removed the links. Phlsph7 (talk) 15:44, 7 August 2024 (UTC)

• On the former, the version that's actually in this article is File:Francois_Viete.jpeg - the one you changed is the version that's in the article that was previously linked (apologies for the confusion). The version in this article still needs tagging and also has a dead source link. Nikkimaria (talk) 01:00, 8 August 2024 (UTC)

  That's confusing indeed, thanks for catching it. I added the US tag and replaced the dead link. Phlsph7 (talk) 07:27, 8 August 2024 (UTC)

  There was a similar problem with Descartes's image. To avoid future confusion, the images currently linked in the article are https://commons.wikimedia.org/wiki/File:Francois_Viete.jpeg and https://commons.wikimedia.org/wiki/File:Frans_Hals_-_Portret_van_Ren%C3%A9_Descartes_(cropped).jpg. Phlsph7 (talk) 07:32, 8 August 2024 (UTC)

UC

I know even less about this one than ethics, so a more sensible person would stay away -- a few comments regardless:

• Linear algebra is a closely related field investigating variables that appear in several linear equations, called a system of linear equations. It tries to discover the values that solve all equations at the same time.: "all equations in the system"? As phrased, it sounds like we mean all equations in existence.

  Good catch. Added. Phlsph7 (talk) 16:56, 7 August 2024 (UTC)

• Algebraic methods were first studied in the ancient period: as with ethics, I think we could do with being a little more precise. Are we happy, for instance, that the Rhind papyrus is the earliest document to concern algebra, and/or that none exists older than the 2nd millennium BCE, or that the oldest known studies are from Egypt?

  We could mention the example of the Rhind Papyrus with a date in this lead paragraph. But I'm not sure that we want to go into those details in the lead. Phlsph7 (talk) 15:45, 9 August 2024 (UTC)

• A matter of taste, maybe, but I'm not sure about a 1980s postage stamp (especially as the likeness is almost certainly totally fictitious) for al-Khwarizmi, especially if we're not going to tell the reader what the image is (some will, I'm sure, assume it's a contemporary portrait). Elsewhere, we have this, which is a page (I think the first page?) from the text we're discussing -- would that be a better illustration here?

  Good idea, I replaced the image, especially since there were also some copyright concerns above. Phlsph7 (talk) 08:41, 7 August 2024 (UTC)

• Transliterated Arabic should be in a transliteration template.

  Done. Phlsph7 (talk) 16:56, 7 August 2024 (UTC)

• A higher level of abstraction is achieved in abstract algebra: is is achieved the right phrase here -- it sounds like we're saying that abstract algebra is better than elementary algebra, when surely they're each trying to do slightly different things? "Abstract algebra uses/creates/allows a higher level of abstraction"?

  Reformulated. Phlsph7 (talk) 16:56, 7 August 2024 (UTC)

• Suggest linking countable noun.

  Added. Phlsph7 (talk) 16:56, 7 August 2024 (UTC)

• Very pedantic, but the link on " a certain type of binary operation" covers the "a", while the one on "a specific type of algebraic structure" doesn't. Usual form is to include it, I think.

  I included the "a" in the wikilink scope. Phlsph7 (talk) 16:56, 7 August 2024 (UTC)

• It's a little odd that note c mentions al-Khwarizmi before we've introduced him, and when we're talking about the use of the term prior to his work. I'd move it to the end of the following sentence.

  Done. Phlsph7 (talk) 16:56, 7 August 2024 (UTC)

• This changed in the course of the 19th century: why not, simply, in the nineteenth century?

  Simplified. Phlsph7 (talk) 16:56, 7 August 2024 (UTC)

• Note e (definition of constants vs. variables) does strictly need a citation, even though it's not exactly controversial. There are a couple of others -- I noticed l and s.

  Done, I hope I got all. I reformulated footnote l to be only about this article. Phlsph7 (talk) 16:56, 7 August 2024 (UTC)

  I'm not going to kick up a huge fuss on this point, but it would be reassuring to provide a citation to show that other people do this too (in other words, that it's not our own idea). UndercoverClassicist ^T·C 17:03, 7 August 2024 (UTC)

  Done. Phlsph7 (talk) 08:01, 8 August 2024 (UTC)

• The lowercase letters a and b are usually used for constants and coefficients. For example, the expression 5x+3 is an algebraic expression created by multiplying the number 5 with the variable and adding the number 3 to the result. I don't really understand the use of "for example" here -- I think we need to introduce this as a new thought.

  Done. Phlsph7 (talk) 08:04, 8 August 2024 (UTC)

• An equation is a statement formed by comparing two expressions with an equals sign ... Inequations are formed with symbols like the less-than sign (), the greater-than sign (), and the inequality sign (). : would it be better to explain this in terms of meaning rather than symbology? After all, the statement "the square on the hypotenuse is equal to the squares on the other two sides" was as much as passed for an equation for a large part of mathematical history, and we can imagine some other form of notation that expresses equations with a different sign or none at all.

  I reformulated it to take a middle path, covering both meaning and symbols. Phlsph7 (talk) 08:47, 8 August 2024 (UTC)

• Some algebraic expressions take the form of statements that relate two expressions to one another: two or more?

  Two is the typical format in algebra. Our formulation leaves it open whether there are other alternatives. Phlsph7 (talk) 08:47, 8 August 2024 (UTC)

  It does. UndercoverClassicist ^T·C 14:40, 9 August 2024 (UTC)

• The main goal of elementary algebra is to determine for which values a statement is true: more idiomatic, to me, as "the values for which a statement is true".

  Reformulated. Phlsph7 (talk) 08:47, 8 August 2024 (UTC)

• In a similar way, if one knows the exact value of one variable one may be able to use it to determine the value of other variables.: do we lose anything important by cutting exact here?

  Not really. Removed. Phlsph7 (talk) 08:47, 8 August 2024 (UTC)

• It bothers me that the graph example uses the wrong symbol for subtraction (the dash is too short), but that's not really your problem. Do consider, however, MOS:COLOUR in describing the line as red (not everyone can see its colour) -- perhaps also add that it slopes upwards to the right? I appreciate that there's only one line that graph-literate readers could identify, but we're rightly pitching this article to complete beginners, and it's not impossible that some won't know what the axes are.

  I updated the image file to use a longer symbol for subtraction (I had to close and re-open my browser for it show the new version). I also followed your suggestion to identify the line not only by color but also by slope. Phlsph7 (talk) 08:47, 8 August 2024 (UTC)

• This means that no variables are multiplied with each other and no powers of variables occur.: or that no variables are raised to a power greater than one?

  Reformulated. Phlsph7 (talk) 08:47, 8 August 2024 (UTC)

• A system of equations that has solutions is called consistent. This is the case if the equations do not contradict each other: better as A system of equations that has solutions is called consistent if the equations do not contradict each other? At the moment, we state something, and then immediately seem to state that it isn't (always) true. If I've got it right, we're saying that it's impossible to be inconsistent and to have any solutions, so it might even be clearer to state that first -- something like If two or more equations contradict each other, the system of equations is called inconsistent and has no solutions. For example, the equations $x_{1}-3x_{2}=0$ and $x_{1}-3x_{2}=7$ contradict each other since no values of $x_{1}$ and $x_{2}$ exist that solve both equations at the same time. If two or more equations contradict each other, the system of equations is inconsistent and has no solutions. A system of equations that has solutions is called consistent.

  I implemented a similar reformulation. Phlsph7 (talk) 12:32, 8 August 2024 (UTC)

• LU Decomposition: decap decomposition, and consider spelling out lower–upper?

  Decap done. I kept the "LU" since this is the more common way of referring to it and also the name of our article. Phlsph7 (talk) 12:32, 8 August 2024 (UTC)

• On a geometric level, systems of equations can be interpreted as geometric figures: do we need the first bit? Seems repetitive, given the end of the sentence.

  Removed. Phlsph7 (talk) 12:32, 8 August 2024 (UTC)

• For systems with two variables, each equation represents a line in two-dimensional space. The point where the two lines intersect is the solution: can we briefly explain why this is so?

  I added a short explanation. Phlsph7 (talk) 12:32, 8 August 2024 (UTC)

• binary operations, which take two objects as input: as inputs, surely, or as their input?

  Done. Phlsph7 (talk) 12:32, 8 August 2024 (UTC)

• The date of the Rhind papyrus -- can we give an idea of the "error bars" on the debate -- does everyone agree it's C17th, for example, or do some people think it's much later, or a modern forgery?

  Done. Phlsph7 (talk) 12:32, 8 August 2024 (UTC)

• These developments happened in the ancient period in diverse regions such as Babylonia, Egypt, Greece, China, and India: our phrasing here implies that this list isn't exhaustive. Is that what's intended?

  Simplified. Phlsph7 (talk) 12:32, 8 August 2024 (UTC)

• In ancient China: I know the date of The Nine Chapters on the Mathematical Art isn't totally straightforward, but we should give it one anyway, if the Greeks get them.

  Added. Phlsph7 (talk) 12:32, 8 August 2024 (UTC)

• On Vector 22, I have quite a lot of sandwiching between the Al-Khwarizmi manuscript and the double portrait. I realise I've earlier suggested using it in place of Al-Khwarizmi's face, so that would solve this problem as well.

  Done. Phlsph7 (talk) 08:42, 7 August 2024 (UTC)

• We get a lot of people described as "the nationality mathematician": you could consider dropping mathematician in these contexts, and take it as read that we're generally talking about mathematicians (see User:Caeciliusinhorto/Context considered harmful for an argument for this). Very much a matter of taste, though.
• the German mathematicians ... Emil Artin -- he was Austrian.

  Done. Phlsph7 (talk) 12:32, 8 August 2024 (UTC)

• to solve puzzles like Sudoku and Rubik's cube: we generally speak of a Rubik's cube, so I'd pluralise it here (especially as there are different variations on the form).

  Done. Phlsph7 (talk) 12:32, 8 August 2024 (UTC)

• Algebra education mostly focuses on elementary algebra, which is one of the reasons why it is also called school algebra. It is usually not introduced -- the series of its may not be totally clear -- we seem to be swapping antecedent here (the first one is elementary algebra, the second algebra education).

  I replaced the first "it". Phlsph7 (talk) 12:32, 8 August 2024 (UTC)

• unlike arithmetic calculations, algebraic expressions often cannot be directly solved: you might add an easily in here -- you can solve most school problems by trial and error, or simply by spotting the answer, but it's much easier to do it "properly".

  I guess it depends on whether we read "often cannot" as "in many cases, there is no logical possibility" or as "in many cases, there is no reliable way to do so". I would add it but "easily directly solved" sounds a bit odd. Phlsph7 (talk) 12:32, 8 August 2024 (UTC)

  To me, often cannot easily be directly solved is good English -- or perhaps "are often difficult to solve directly"? UndercoverClassicist ^T·C 14:06, 8 August 2024 (UTC)

  Done. Phlsph7 (talk) 16:03, 8 August 2024 (UTC)

• On the education side -- how worldwide is the use of balance scales? We've cited a single school textbook here, which to me is verging on WP:PRIMARY -- I'd be more comfortable with a survey article on the use of the technique in (American?) mathematical education. I've never seen it in UK schools, outside isolated word problems -- function machines are more common over here as a basic introduction to the ideas of algebra. I'd be interested to know how things are done in places like Shanghai and Hong Kong, which generally seem to outperform both systems, at least as far as concerns producing students who can solve school-algebra problems.

  I added two more sources that that examine some research on this approach. Phlsph7 (talk) 12:32, 8 August 2024 (UTC)

  Both of these sources (particularly Kaput 2018) consider balance scales as one tool among many -- Kieran talks about other manipulatives (cups and sticks, for example), and Kaput talks about a whole bunch of visualisations (including function machines), particularly those which they feel to be appropriate for use with a computer. I think the discussion here needs to be broadened to reflect those sources -- there's a good point to be made that teaching algebra often involves using conceptual tools, often ones with which students can interact physically, before introducing abstract concepts such as variables, but we shouldn't frame that entirely through one of those tools (balance scales). There's an interesting booklet for teachers here, with extensive bibliography, which recommends the use of representations but also acknowledges that the evidence for their effectiveness (like everything else in education!) is minimal. UndercoverClassicist ^T·C 14:13, 8 August 2024 (UTC)

  I added some additional context to the discussion by mentioning manipulatives and visualizations. Phlsph7 (talk) 15:41, 9 August 2024 (UTC)

  I've made a small and slightly pedantic fix (computers aren't visualisations), but I think this works well now. Feel free to counter-tweak. UndercoverClassicist ^T·C 16:37, 9 August 2024 (UTC)

  Looks good. Phlsph7 (talk) 16:48, 9 August 2024 (UTC)

Can I echo Roy's praise for the clarity and approachability of this article -- I'll admit that I skipped fairly lightly over the abstract algebra section, but the rest was absolutely clear and manageable, and I suspect I'm going to be one of the least qualified mathematicians to review this here. Excellent work once again. UndercoverClassicist ^T·C 06:55, 7 August 2024 (UTC)

Thanks for yet another detailed review and for taking a leap to provide a non-expert opinion on the article! Given that Wikipedia is a general encyclopedia, this is also an important perspective to consider. Phlsph7 (talk) 08:55, 7 August 2024 (UTC)

Support: I am hugely impressed by the writing and clarity here, and while I am not qualified to vouch for the mathematics, everything within my expertise looks excellent. UndercoverClassicist ^T·C 16:38, 9 August 2024 (UTC)

Thank you for the support! Phlsph7 (talk) 08:39, 10 August 2024 (UTC)

Comments Edwininlondon

Although I am neither an expert in the field nor a native speaker, I have a few comments. Overall I very much appreciated the clarity and structure.

• Algebra is the branch of mathematics that studies algebraic structures --> while I like overall how clearly topics are being described, there are a few cases where I get a sense of recursion. Is there a way to avoid using algebraic in the definition of algebra?

  You are right that this sounds circular, but I'm not sure that there is a good alternative. There was already a detailed discussion on this point in the GA review that resulted in consensus on the current formulation. It sounds circular but it isn't circular since the technical term "algebraic structure" is defined without reference to algebra. Phlsph7 (talk) 16:05, 9 August 2024 (UTC)

• called a system of linear equations. --> why italics?

  This is per MOS:WORDSASWORDS since we talk about the term "system of linear equations". Phlsph7 (talk) 16:05, 9 August 2024 (UTC)

• that appear in several linear equations --> a definition of linear would be good

  We could add a footnote but I'm not sure that we should get into this in the lead section. The first paragraph of the subsection "Linear algebra" provides a definition. Phlsph7 (talk) 16:05, 9 August 2024 (UTC)

• Algebra is the branch of mathematics that studies algebraic operations[a] and algebraic structures --> too recursive for me. A question: do I understand it correctly that algebraic structures include lgebraic operations? If so, do we really need to say "algebraic operations[a] and algebraic structures" or can it just use structures?

  You are right, the operations are already included. I found a way to reformulate the sentence to take this into account while cutting down on the repetitive language. Phlsph7 (talk) 16:05, 9 August 2024 (UTC)

• Arithmetic studies arithmetic operations --> too recursive for me

  I removed the term "arithmetic" since we already list the main operations. Phlsph7 (talk) 16:05, 9 August 2024 (UTC)

• first use of axiom in body is not linked. Perhaps do this: together with their underlying axioms, the laws they follow.

  Done. Phlsph7 (talk) 16:05, 9 August 2024 (UTC)

• The natural numbers, by contrast, do not form a group --> should the + operation not be mentioned?

  Correct, I added it. Phlsph7 (talk) 16:35, 9 August 2024 (UTC)

• One of the earliest documents is the Rhind Papyrus --> One of the earliest mathematical documents is the Rhind Papyrus

  I implemented the idea in a slightly different way. Phlsph7 (talk) 16:35, 9 August 2024 (UTC)

• several generations between the 10th century BCE and the 2nd century CE --> I don't think 1000+ years can be described as several generations

  Agreed, this is an understatement. I adjusted the text. Phlsph7 (talk) 16:35, 9 August 2024 (UTC)

• Thābit ibn Qurra in the 9th century --> I assume there is no more accurate estimate? Since 825 CE is also the 9th century, perhaps something like "also in the 9th century"?

  I think he made contributions in several works so we would have to list several dates. I added the "also". Phlsph7 (talk) 16:35, 9 August 2024 (UTC)

• In 1247, the Chinese mathematician Qin Jiushao --> how come this is not with the other China info?

  For chronological reasons: roughly speaking, we have two paragraphs on ancient history, two paragraphs on post-classical history, and then modern history. It's not ideal but putting him into the ancient paragraph is not ideal either. Phlsph7 (talk) 16:35, 9 August 2024 (UTC)

That's all I could see. Nice work. Edwininlondon (talk) 19:50, 8 August 2024 (UTC)

Hello Edwininlondon, I appreciate you taking the time to review this article! Phlsph7 (talk) 16:05, 9 August 2024 (UTC)

A bit more:

• x y c – variables/constants --> would be better to make 2 lines and separate the variables from the constant

  Done. Phlsph7 (talk) 11:12, 12 August 2024 (UTC)

• forgive me if this has been agreed already, but is Encyclopedia of Mathematics as a wiki a reliable source?

  Thanks for raising this point. The website use wiki software to display the pages but it is not user-generated. The articles were originally published in book form by Kluwer Academic Publishers/Springer and only later made accessible online. We could cite the original books but the online version is much better accessible for readers. Phlsph7 (talk) 11:12, 12 August 2024 (UTC)

• inconsistent date formats. Example: "from the original on 4 October 2009. Retrieved 23 October 2023" but also "from the original on 2024-01-12. Retrieved 2024-01-13"

  Done. I hope the script got all. Phlsph7 (talk) 11:41, 12 August 2024 (UTC)

• Walz, Guido (2016) needs a trans-title

  Added. Phlsph7 (talk) 11:41, 12 August 2024 (UTC)

• Spotcheck: 37 51 84 85 87 118 all check out
• 80 does indeed give 1550 but somehow I feel this source is not right to make the claim "The exact date is disputed and some historians suggest a later date around 1550 BCE" A more scientific source would be better.

  I replaced it with a better source. Phlsph7 (talk) 11:41, 12 August 2024 (UTC)

• 87 is correct but the link unfortunately does not put me on page 31, nor is that page accessible to me. Perhaps this link is better: https://nap.nationalacademies.org/read/11540/chapter/4

  I fixed the link, the page preview works for me now. Phlsph7 (talk) 11:41, 12 August 2024 (UTC)

That's it for this final round. Edwininlondon (talk) 07:16, 11 August 2024 (UTC)

Thanks for your comments. I hope I was able to address the main concerns. Phlsph7 (talk) 11:41, 12 August 2024 (UTC)

Yes. Almost all fine, except that there still is a lingering sadness in me regarding the opening sentence with its circularity. I don't think the argument that the technical term "algebraic structure" is defined without reference to algebra is particularly strong. But I lack the expertise to provide something useful. Perhaps it is something like "Algebra is ..., known as algebraic structures, ... I was thinking perhaps the part "manipulation of statements within those structures" can be dropped, as that surely is encompassed by the word "studies". But maybe the phrase "manipulation of statements" is rather critical, as it conveys the essence of the field. Sorry, I can't express what is better. Edwininlondon (talk) 12:31, 16 August 2024 (UTC)

The current first sentence is: "Algebra is the branch of mathematics that studies algebraic structures and the manipulation of statements within those structures". This definition is not circular since "algebraic structure" has a precise definition that does not refer to the field of algebra. So it's not a problem with the definition itself but only with the linguistic level since it is preferable to avoid repeating the words algebra-algebraic.

I'll brainstorm some alternatives:

1. Algebra is the branch of mathematics that studies abstract structures and the manipulation of statements within those structures

   The expression "abstract structures" does not have a precise definition and could mean all kinds of things, so this formulation sacrifices information for linguistic improvements

2. Algebra is the branch of mathematics that studies operations from a Cartesian power of a set into that set and the manipulation of statements using these operations.

   This is precise but most readers will have difficulties figuring out what "operations from a Cartesian power of a set into that set" means. Especially for the first sentence, this is not a good idea.

3. Algebra is the branch of mathematics that studies operations on mathematical objects and the manipulation of statements using these operations.

   This is a less detailed and more accessible version of (2). Instead of repeating algebra-algebraic, this formulation repeats mathematics-mathematical.

When compared to these alternatives, I prefer the current version, but I'm also open to other ideas. Option 3 would be my second choice.

Roughly speaking, the first clause on algebraic structures covers abstract/universal algebra while the second clause on the manipulation of statements covers elementary/linear algebra. If we removed the second clause, we would focus only on the more abstract side of algebra. Phlsph7 (talk) 07:25, 18 August 2024 (UTC)

Thanks for generating alternatives, much appreciated. I'd be curious to hear what other FAC reviewers think. In my mini-sample of 2 non-maths people, both raised an eyebrow at "algebraic". Alternative 2 is too technical indeed. Number 3 would be my preferred option. Edwininlondon (talk) 12:24, 18 August 2024 (UTC)

Any mileage in "certain abstract structures, known as algebraic structures", or similar? UndercoverClassicist ^T·C 13:47, 18 August 2024 (UTC)

4. Algebra is the branch of mathematics that studies certain abstract structures, known as algebraic structures, and the manipulation of statements within those structures.

5. Algebra is the branch of mathematics that studies certain abstract systems, known as algebraic structures, and the manipulation of statements within those systems.

6. Algebra is the branch of mathematics that studies certain abstract frameworks, known as algebraic structures, and the manipulation of statements within those frameworks.

All of them are a little bit longer than the original. Maybe they could work without the word "certain". In (4), the repeated use of the word "structure" might be a problem. Of these three, (5) would be my preference. Phlsph7 (talk) 15:31, 18 August 2024 (UTC)

I like (5). Edwininlondon (talk) 08:50, 19 August 2024 (UTC)

5 makes sense to me as a layman, though obviously I can't speak for its technical accuracy/completeness. UndercoverClassicist ^T·C 16:15, 19 August 2024 (UTC)

I implemented the suggestion. It's a little longer but should be more accessible. Phlsph7 (talk) 07:40, 20 August 2024 (UTC)

I Support on prose. My uni algebra is too long ago to fully vouch for the technical aspect, but it looks very convincing. A nice piece of work. Edwininlondon (talk) 17:00, 20 August 2024 (UTC)

Thanks for your helpful comments and the support! Phlsph7 (talk) 07:17, 21 August 2024 (UTC)

Mathwriter2718's comments

I am here in my capacity as a mathematical expert. My goal is to review the mathematical content of this page and make sure it is accurate and clear. Others have already done more comprehensive reviews of other aspects.

I think this article is excellent. I have a few extremely minor concerns and one bigger concern, but none will take so long to address. For the big concern, Let us look at the descriptions of linear algebra, abstract algebra, and universal algebra in this article.

"Linear algebra is a closely related field investigating variables that appear in several linear equations, called a system of linear equations. It tries to discover the values that solve all equations in the system at the same time. Abstract algebra studies algebraic structures, which consist of a set of mathematical objects together with one or several binary operations defined on that set. It is a generalization of elementary and linear algebra since it allows mathematical objects other than numbers and non-arithmetic operations. ... Universal algebra constitutes a further level of generalization that is not limited to binary operations and investigates more abstract patterns that characterize different classes of algebraic structures."

"Abstract algebra usually restricts itself to binary operations that take any two objects from the underlying set as inputs and map them to another object from this set as output."

"Universal algebra is the study of algebraic structures in general. It is a generalization of abstract algebra that is not limited to binary operations and allows operations with more inputs as well, such as ternary operations."

I think many mathematicians define linear algebra as the study of finite-dimensional vector spaces. The description of linear algebra in this article is pretty different on the surface, but still a valid POV, and not actually as different as it may appear. Anyway, it would be nice to put in somewhere that the algebraic structure linear algebra studies is a finite-dimensional vector space. The bigger issue is that everyone thinks vector spaces are under the domain of abstract algebra, and scalar multiplication is not a binary operation on a single set, so the descriptions of abstract algebra and universal algebra are wrong. Even if you expanded abstract algebra to be about binary operations where the input sets can be different, this would still not be how mathematicians view abstract algebra.

I think the way mathematicians view abstract algebra vs universal algebra vs linear algebra is like this:

Abstract algebra is the broad field of math that studies algebraic structures.

Linear algebra is the study of a specific algebraic structure that is important in the study of systems of linear equations: finite-dimensional vector spaces.

Universal algebra is the study of a specific algebraic structure called a universal algebra. This structure is kind of unusual in that its instantiations include many of the most important algebraic structures.

I think we should just remove the offending content and not change things too much otherwise. I am merely arguing that we should avoid explicitly limiting "abstract algebra" to binary operations on a single set, and that we should avoid thinking of universal algebra as a generalization of abstract algebra, but rather as the study of a structure that encases many of the most important algebraic structures. If there are no objections, I can make these changes.

@Mathwriter2718: Thanks for taking a look at the article! I followed your suggestion to mention that linear algebra can also be defined in terms of vector spaces. I included the reference to linear maps in the definition so it is more focused. I put it in a footnote since I have the impression that it is difficult to understand for the average reader but we could try to work it into the main text if that is preferable.

Concerning abstract algebra, one problem is that some sources restrict abstract algebra to binary operations. In order to avoid taking sides, I softened this claim by saying that it is "primarily interested in binary operations".

The relation between abstract and universal algebra is tricky. Pratt 2022 says "Universal algebra is the next level of abstraction after abstract algebra". Other sources also emphasize the general nature of universal algebra but don't make the relation to abstract algebra this explicit. I reformulated some passages to emphasize the generality. I tried not to imply that universal algebra is distinct from and more general than abstract algebra. I also added a footnote covering the alternative definition of universal algebra as the study of universal algebras, as you suggested.

I hope these changes are roughly what you had in mind. Phlsph7 (talk) 12:25, 30 August 2024 (UTC)

Seems like a good compromise to me. Thanks. Mathwriter2718 (talk) 12:44, 30 August 2024 (UTC)

Less important comments:

• Consider replacing

  For example, the expression 7x − 3x can be replaced with the expression 4x.

with

For example, the expression 7x − 3x can be replaced with the expression 4x, since 7x - 3x = (7-3)x = 4x by the distributive property.

Done. Phlsph7 (talk) 12:25, 30 August 2024 (UTC)

• Consider replacing

  This technique is common for polynomials to determine for which values the expression is zero.

with

This technique is commonly used to determine the values of a polynomial that evaluate to zero.

Done. Phlsph7 (talk) 12:25, 30 August 2024 (UTC)

• The article uses the term "x-y pair". I don't think I've seen this before, but it perhaps isn't wrong? I would expect to see (x,y)-pair or (x,y) pair, I think.

  Done. Phlsph7 (talk) 12:25, 30 August 2024 (UTC)

• Maybe it could be good to put in some image of a symmetry group when the article is talking about symmetry groups, though I didn't find an image on dihedral group that really stood out to me.

Mathwriter2718 (talk) 13:20, 29 August 2024 (UTC)

After a short look, I didn't find a good image either. This part of the article already has several images so we might have to remove an image to create space for a new one. Phlsph7 (talk) 12:25, 30 August 2024 (UTC)

@Mathwriter2718: I appreciate the insightful suggestions. I hope I was able to address your main concerns. I was wondering whether, from the mathematical perspective, you would support the nomination. Phlsph7 (talk) 07:33, 2 September 2024 (UTC)

@Phlsph7 I hope that I can read up more on universal algebra before giving an answer. This might take a bit. Mathwriter2718 (talk) 14:30, 4 September 2024 (UTC)

Thanks for taking another look at the subsection "Universal algebra". The main challenge for this subsection is to make the abstract topic accessible to the reader without oversimplifying too much. Phlsph7 (talk) 08:06, 5 September 2024 (UTC)

Hi Phlsph7, is this ready for the reviewer to take another look at yet? Gog the Mild (talk) 19:01, 15 September 2024 (UTC)

@Gog the Mild: The article is ready and, as far as I'm aware, there are no outstanding comments to be addressed. Mathwriter2718 said that they needed more time to familiarize themselves with the literature before wrapping up the review.

@Mathwriter2718: Just checking to see how things are progressing. Please let me know if I can be of any assistance. Phlsph7 (talk) 07:54, 16 September 2024 (UTC)

@Phlsph7 I am very sorry for not getting back to you sooner!! I have recently become extremely busy and I have had trouble finding the time to review the mathematical literature and decide whether I support/don't support this nomination. I will give myself a deadline of tonight to finish this and if I can't get it done by then, then I think I can declare I just don't have enough time right now to do this. Mathwriter2718 (talk) 14:17, 25 September 2024 (UTC)

@Mathwriter2718: Thanks for taking another look! If turns out that you don't have the time to review the part on universal algebra, you could explicitly exclude that part from your assessment. Phlsph7 (talk) 16:33, 25 September 2024 (UTC)

@Phlsph7 @Gog the Mild I have finished my review of the mathematical content of the article except for the "Universal algebra" section and I support the FA nomination based on the content that I have reviewed. Maybe there is someone else who can review that section but I am not sure if there are many Wikipedians familiar with universal algebra. Looking at the history of the page Universal algebra one can maybe find people who are familiar with the subject. @Jochen Burghardt has a decent number of edits there. Personally, I'm just not qualified to offer my perspective on that area and I am too busy at this time to really become familiar in the way I would like to before offering an opinion. I'm sorry I couldn't be of more assistance here. Mathwriter2718 (talk) 03:14, 26 September 2024 (UTC)

Thank you for the support and all the time and energy you have poured into this review! Phlsph7 (talk) 07:32, 26 September 2024 (UTC)

Source review

So, this is one of these broad topics where it's hard to tell for an outsider whether the coverage is representative. So I'll qualify that I am not reviewing that aspect of a source review. I wonder why some page numbers have Google Books links and others don't. Google Books serves up different results to different people, so I am not sure that these links are very helpful at all. By the same principle, I don't think that Google Books needs archive links. Springer is referred to by various names, is there a need for consistency? Are Jones & Bartlett Publishers and Linus Learning a prominent publisher? What makes "Edwards, C. H. (2012). Advanced Calculus of Several Variables. Courier Corporation. ISBN 978-0-486-13195-5. Archived from the original on January 24, 2024. Retrieved January 24, 2024." a high-quality reliable source? "Majewski, Miroslaw (2004). MuPAD Pro Computing Essentials (2 ed.). Springer. ISBN 978-3-540-21943-9.", "Nicholson, W. Keith (2012). Introduction to Abstract Algebra. John Wiley & Sons. ISBN 978-1-118-13535-8." and "Mishra, Sanjay (2016). Fundamentals of Mathematics: Algebra. Pearson India. ISBN 978-93-325-5891-5." don't have the retrieval dates where other sources have, although with books and papers I don't think we need these at all. Otherwise we are using prominent publishers and series, although I notice the overrepresentation of Western sources. Jo-Jo Eumerus (talk) 08:21, 4 September 2024 (UTC)

Hello Jo-Jo Eumerus, thanks for taking care of the source review! I usually add links to google book pages that offer page previews if I'm aware of them. For some books, google books does not offer previews, in which case I can't add links. It could depend on the reader's geo-region whether a page preview is available. If it is available, it is a convenient way for the reader to verify the material without needing to buy the book. I removed all the google book webarchive links. The problem is that IABot adds them automatically when it runs, so they could be back soon. I implemented a more consistent approach for referring to Springer. I replaced the sources by Jones & Bartlett Publishers, Linus Learning, and Edwards 2012 with alternatives. I added an access/retrieval date for Nicholson 2012. Majewski 2004 and Mishra 2016 don't have access dates because they have no links to a website. The overrepresentation of sources by Western publishers in the article reflects the general prevalence of Western publishers regarding high-quality English-language sources on the subject.

Phlsph7 (talk) 17:02, 4 September 2024 (UTC)

Seems to me like we need some discussion somewhere about IAbot adding archives to Google Books. But not an issue for a FAC I figure. Jo-Jo Eumerus (talk) 06:18, 5 September 2024 (UTC)

Right, this has already come up several times. I started a discussion at Wikipedia_talk:Featured_article_candidates#Google_Books_web_archive_links_and_IABot. Phlsph7 (talk) 08:04, 5 September 2024 (UTC)

• Hi Jo-Jo, sorry to drag you back to this again. Would I be correct in understanding this to be a source review pass so far as FAC is concerned? Thanks. Gog the Mild (talk) 15:26, 26 September 2024 (UTC)

  Yes, but note the caveats "So I'll qualify that I am not reviewing [the thorough and representative survey] aspect of a source review." and "overrepresentation of Western sources." I am not sure that limiting oneself to English sources justifies incompleteness, although we can't expect editors to be polylingual. Jo-Jo Eumerus (talk) 06:26, 27 September 2024 (UTC)

D.Lazard

I did not follow the changes of the article done by Phlsph7 since January 2023. My first impression is that the new vesion is much better. Nevertheless it is too much biased toward educational aspects of algebra. I'll discuss this in several items in order to makes improvements easier.

• Abstract algebra: Presently, this phrase is almost never used outside mathematical education. This must be said in the article, and in many occurences of this phrase the word "algebraabstract" must be removed.

  Hello D.Lazard and thanks for taking a look at the article! I found a source that talks about how the term "abstract algebra" is used in the educational context and added a sentence on it. I'm not sure if this is what you had in mind. If you know of a source that spells your point out in more detail, I would be happy to have a look at it. Most of the sources that I'm aware of define abstract algebra as a field of inquiry rather than a math course in undergraduate studies.

  Clearly, I have no source saying "I do not use abstract algebra because this is reserved to educational context", but, AFAIK, there is no recent sources (say, not older than 50 years) that use "abstract algebra" outside educational or historical context. D.Lazard (talk) 16:59, 26 September 2024 (UTC)

  You seem to suggest the addition of a sentence like the following: "In the last 50 years, the term abstract algebra has only been used in educational and historical contexts." I'm not opposed in principle, but we would need to figure out how to source this sentence and how to deal with possible counterexamples like [2], [3] and [4]. Phlsph7 (talk) 08:26, 27 September 2024 (UTC)

  I didn't get what you mean by in many occurences of this phrase the word "algebra" must be removed. Phlsph7 (talk) 16:18, 26 September 2024 (UTC)

  Of course, it was a typo, I meant "abstract". D.Lazard (talk) 16:59, 26 September 2024 (UTC)

  Ah, I see. Depending on the context, the term algebra is sometimes used to refer only to elementary algebra or only to abstract algebra, as explained in our section "Definition and etymology". I think it's in the best interest of the readers to use the more specific names to avoid confusing them. Phlsph7 (talk) 08:31, 27 September 2024 (UTC)

  Sorry, reading the article again, I do not find any improper use of "abstract algebra", except in the last paragraph of section "abstract algebra", which is controversial for other reasons, for example, by asserting implicitly that, say, the the study of groups of geometric transformations belong to abstract algebra. D.Lazard (talk) 10:21, 27 September 2024 (UTC)

  Done. Phlsph7 (talk) 15:39, 27 September 2024 (UTC)

• Major branches: Presently, this section has only 4 subsections: "Elementary algebra", "Linear algebra", "Abstract algebra", "Universal algebra".
  For defining branches of algebra, the most authoritative source is the Mathematics Subject Classification. The previous version of the article referred to this source by writing: Today algebra includes section 08-General algebraic systems, 12-Field theory and polynomials, 13-Commutative algebra, 15-Linear and multilinear algebra; matrix theory, 16-Associative rings and algebras, 17-Nonassociative rings and algebras, 18-Category theory; homological algebra, 19-K-theory and 20-Group theory. Algebra is also used extensively in 11-Number theory and 14-Algebraic geometry.
  Three of the four major areas belong to 08-General algebraic systems, and very little is said about the seven other major areas of algebra. So, without sections on other major branches of areas, this article fails the second criterion of featured articles: comprehensive: it neglects no major facts or details and places the subject in context

  As far as I'm aware, the Mathematics Subject Classification does not provide a general definition or subdivision of algebra. The sentences you cited from an older version of our article were unsourced. If you know of a source that supports this subdivision of algebra, I would be interested to read it. The sources that I'm aware of do not divide it this way, but there may be different ways of dividing it.

  Most of the categories you mention are covered in one form or another in our article. For example, polynomials are covered by the section "Linear algebra", algebraic geometry and algebraic number theory are mentioned in the subsection "Other branches of mathematics", and the different rings, fields, and algebras would be belong to the section "Abstract algebra". The last paragraph of the section "Abstract algebra" gives various examples of different algebraic structures. We could add more if you feel that this would help comprehensiveness, but we probably shouldn't overdo it. Phlsph7 (talk) 16:48, 26 September 2024 (UTC)

  As Mathematics Subject Classification was elaborated by the whole mathematical community, this is a reliable source for subdivision of mathematics; it is undoubtly much more reliable than any text written by a single author. So, it is wrong to say that the old paragraph was unsourced. Per WP:BLUE, no source is needed for asserting that field theory, polynomial theory, commutative algebra, associative rings and algebras, nonassociative rings, homological algebra, and group theory are branches of algebra. Nevertheless it suffices to open any textbook having these subjects in their title to see that these subjects belong to algebra. So, the present state of the article breaks policy WP:NPOV by giving much less place to all these subjects together than to universal algebra, a subject that is not really used outside itself (it has been replaced by the much more powerful category theory). D.Lazard (talk) 21:31, 26 September 2024 (UTC)

  I have my doubts that this is a WP:BLUE-statement. The Mathematics Subject Classification is a reliable source but does not support the claim about how algebra is divided into the main branches. Of the items you mentioned, our article provides a detailed explanation of the basics of groups, rings, fields, and polynomials. It also mentions some of the more specific algebras without going too much into detail. I think it's not the responsibility of this type of overview article to go into the more advanced details of these subjects.

  Universal algebra is usually given more weight in reliable sources than the other areas you mentioned so I don't think this violates WP:NPOV. For example, Pratt 2022 discusses it as one of the main branches, without characterizing any of the other areas you mentioned as main branches. Bronshtein's "Handbook of Mathematics" paints a similar picture: in its division "5 Algebra and Discrete Mathematics", universal algebra gets the main subdivision "5.3 Univeral algebra" but none of the other fields you mention get main subdivisions. But you are right that universal algebra is not as important as abstract algebra. I could try to reduce the length of the section "Univeral algebra" by boiling it down to 2 paragraphs. Would that address your main concern? I was thinking about adding a few sentences on homological algebra but this topic could be quite challenging to the average reader. Phlsph7 (talk) 15:30, 27 September 2024 (UTC)

  The most influential graduate textbooks in algebra are probably Van der Waerden's Algebra and Lang's Algebra. They are probably the most reliable sources for this subjects. In particular, Lang has more than 10,000 citations. It is unbelievable that none of these books is cited in our article. Clearly, they do not contain everything that belongs to algebra, but they are certainly reliable sources for the important branches of algebra. I have not these books under hand, but the phrases abstract algebra and universal algebra do not appear in the table of content of Lang nor in the list of Van der Waerden's chapters. On the other hand, all the branches that are cited above appear, at least in Lang (which is more recent). So, the fact that the above cited branches belong to algebra is supported by reliable sources. They support also the fact that "abstact algebra" is not used outside educational level, and that universal algebra is not a major branch of algebra (Lang knew universal algebra, since he is an author of an article on this subject, and did not included the subkect in his book). D.Lazard (talk) 18:06, 27 September 2024 (UTC)

• Sections "Abstract algebra" and "Universal algebra" are both devoted to the general study of algebraic structures. So, there must be merged and shortened for making place to (presently lacking) other major branches that are much more active.
• Theorems: The article is misleading by suggesting wrongly that there are no important theorems in algebra. As an example, one thinks immediatly of Feit–Thompson theorem, which is the theorem of algebra with the longest proof (the complete proof of Wiles' proof of Fermat's Last Theorem is probably longer, and contains much algebra, but is not limited to this area of mathematics). Other examples are Hilbert's basis theorem, Hilbert's Nullstellensatz and Hilbert's syzygy theorem. Beside their historical importance, they are interesting here, since they predate "abstract algebra", and are therefore difficult to classify in this branch of algebra.

  I responded to this and the two preceding points on the article talk page, which is probably a better place for this type of discussion. Phlsph7 (talk) 07:55, 28 September 2024 (UTC)

@D.Lazard It seems to me like your overarching concern is that the article Algebra should be about exactly the thing mathematicians call "algebra". I think it makes more sense for this particular article to be a middle ground between what most people call algebra and what mathematicians mean when they say algebra. For example, you want to remove the use of the word "abstract algebra". Indeed, mathematicians don't use this word to talk to each other about research-level mathematics. But I think it makes a lot of sense to use this word for contrast with elementary algebra. The way I interpret the current state of this article, everything under the mathematics subject classification for algebra falls under what this article calls "abstract algebra".

I probably support your view that this article should be closer to the viewpoint of a modern mathematician than it is now, but I want to be very careful to not transform the intended readership of this article away from the most lay audience possible by going into things like K-theory and nonassociative rings that are certainly not necessary for a comprehensive description of "Algebra" for the lay reader. Math Wikipedia already has a bad reputation for being too technical and obscure, and only a vanishing fraction of those who search for the article Algebra on Wikipedia will have much background. Mathwriter2718 (talk) 00:48, 30 September 2024 (UTC)

The purpose of the Mathematics Subject Classification is to be used by journals to organize research. I don't we should expect that purpose to align very well with what subfields an expository article about "Algebra" should cover. Mathwriter2718 (talk) 00:54, 30 September 2024 (UTC)

The purpose of an encyclopedic article is not to adhere to the conception of some readers ("what most people call algebra"); it is to inform the reader on the whole subject implied by the title. Here, this includes "abstract algebra", and also all the content of the most influencal books entitled Algebra (in particular, Serge Lang's and Van der Waerden's ones). Since no way is given to the reader to accede to information on most of the content of these books, the article is far to respect the policy WP:NPOV, and thus should never to have been labeled as a WP:GA.

I never asked to not speak of "abstract algebra", but it must be given its WP:due weight, which is the name of the part of algebra that is taught at some level of mathematical education. D.Lazard (talk) 10:52, 30 September 2024 (UTC)

I tend to agree with Mathwriter2718 here. Writing an article that respects both the mathematicians' definition of algebra and the more vernacular meaning is an intrinsically difficult problem. Having no representation for the latter meaning would itself be a violation of NPOV, and would make the page far less useful for a large and important audience. Moreover, I agree that the Mathematics Subject Classification doesn't necessarily align very well with what subfields an expository article about "Algebra" should cover. The topics that it lists are important enough to include, but it doesn't dictate the organization of an encyclopedia article. XOR'easter (talk) 16:05, 30 September 2024 (UTC)

I have opened a related discussion at WT:WPM#Should Algebra be reverted to the version of 21 Decembre 2023?. D.Lazard (talk) 15:45, 30 September 2024 (UTC)

D. Eppstein

A drive-by comment: the claim in the universal algebra section that "Two algebraic structures that share all their identities are said to belong to the same variety." and the examples that follow this claim do not match my understanding of the subject. As I understand it, and as Variety (universal algebra) describes, a variety is defined by any set of identities, and an algebra belongs to a variety when it obeys all those identities (even when it might also obey others). So a single algebra might belong to many varieties, not merely the single variety defined by all its identities. Two algebras might belong to one variety, and differ in their membership of another variety. In this same section, "the ring of polynomials" is ambiguous: polynomials over what domain? Footnote [74] appears off-topic; neither linked reference page is about membership of integers, polynomial rings, or rationals in varieties. (One of the two pages uses "variety" in a different sense, from algebraic geometry rather than universal algebra.) The claim that the integers and ring of polynomials (over whatever domain) obey the same identities is unsourced, and may be false depending on the domain of the polynomials. For instance polynomials over GF(2) obey the identity x+x=0 that the integers do not.

Hello David Eppstein and thanks for your comments! I had a look at a few sources and I think your interpretation of varieties is correct. I reformulated the passage to avoid the misleading formulation used earlier. I added the sources I consulted and replaced the example with another. It's a simplified version of the one found in Rosen 2012. If this is still controversial, we could either use the full example from Rosen 2012 or leave it out. Phlsph7 (talk) 11:45, 27 September 2024 (UTC)

There another issue in this section: it is not said explicitely that not all algebraic structure belong to a variety. For example, fields do not form a variety since division by zero is not defined. D.Lazard (talk) 14:08, 27 September 2024 (UTC)

I added a footnote to mention this. Phlsph7 (talk) 08:10, 28 September 2024 (UTC)

XOR'easter

A drive-by comment: this article is well on its way to being very nice indeed. I am somewhat dissatisfied by the "Applications" section. Despite a recent reorganization for the better, it still suffers from the problem that the applications of algebra are so staggeringly vast that any selection of examples will look weird and arbitrary. Right now, it reads rather like, "Linear algebra is useful in optimizing the yield of pumpkin patches." It needs both to be augmented with further examples and to be phrased in a way that makes clear the choice of examples is illustrative, not exhaustive. The "Education" section is also a bit shallow. It makes sense to focus on "elementary" or "school" algebra, as the text currently does, but we should say at least a little about the teaching of "higher"/"modern"/"abstract" algebra at the university level.

The references need to be combed through for cruft. I have removed a couple that looked like someone just pulled the first item in a Google Books search that wasn't obvious crankery [5][6]. XOR'easter (talk) 22:28, 8 October 2024 (UTC)

@XOR'easter: Thanks for your comments. I added a short paragraph on tertiary education to the "Education" sections. I think it's a good idea to mention it but I don't want to delve too much into it given that most of the literature on algebra education focuses on secondary education. I reformulated some parts of the "Applications" section to make sure that the examples are not presented as an exhaustive account. I also expanded it by adding several new examples. If you have specific examples in mind that are not yet mentioned, I can also try to include them. I also replaced some sources with better alternatives. Phlsph7 (talk) 12:49, 10 October 2024 (UTC)

Comments from BorgQueen

@Phlsph7: Nice work, another comprehensive article on a core topic from you! Just one thing—as per MOS:SEEALSO, Editors should provide a brief annotation when a link's relevance is not immediately apparent, when the meaning of the term may not be generally known, or when the term is ambiguous. I notice that you've got quite a few links in the See also section. Are they all really necessary? If so, I suggest you use {{annotated link}} for the links that aren't obviously self-explanatory. BorgQueen (talk) 17:40, 22 October 2024 (UTC)

Hello BorgQueen and thanks for the comments. I'm not sure that all the see-also links are necessary. There were some disagreements about what the article should and shouldn't cover. As a result, many of the more specific topics were added as see-also links. I trimmed the links a little and followed your idea to use annotated links since it wouldn't be clear to the average reader what they all mean. Phlsph7 (talk) 11:44, 23 October 2024 (UTC)

• Closing note: This candidate has been promoted, but there may be a delay in bot processing of the close. Please see WP:FAC/ar, and leave the {{featured article candidates}} template in place on the talk page until the bot goes through. Gog the Mild (talk) 20:08, 28 October 2024 (UTC)

The above discussion is preserved as an archive. Please do not modify it. No further edits should be made to this page.