Talk:Algebra

Changes to the article

I was thinking about implementing changes to this article with the hope of moving it in the direction of GA status. There is still a lot to do. Major parts lack references, and there are a few maintenance tags, such as "citation needed" and "better source needed". The section "Areas of mathematics with the word algebra in their name" is dubious since it is mainly a collection of links that were selected by whether they contain the string "algebra". Maybe this could be converted into a standalone list article but it probably does not meet the notability guidelines, see WP:LISTN. Some of the topics in it are quite relevant (like linear algebra and universal algebra) and should probably be discussed somewhere in the prose of another section rather than in the form of disconnected list items. The most important remaining links could be moved to the "See also" section.

I'm also not sure that it's a good idea to dedicate the whole section "Different meanings of 'algebra'" to disambiguation since this type of discussion belongs more to dictionary entries than to encyclopedia entries (see WP:NOTDICT). The better solution would probably be to convert this section into a Definition section that explains the main meaning of the topic of the article (roughly as a branch of mathematics that covers areas like elementary algebra, linear algebra, abstract algebra, and universal algebra) and then clarifies to the reader that there are also other ways how the term is used (for example, for specific subareas or for specific algebraic structures). This way, we have the disambiguation and hopefully do not violate WP:NOTDICT.

It might further be a good idea to expand the article in a few directions (while still staying within the limits of WP:SIZERULE). Several high-quality overview sources I checked present linear algebra and universal algebra as major subfields of algebra (for example, [1], [2], [3], and the algebra entries of [4] and [5]). The discussion of the application of algebra to other fields should probably also get some more details, like the algebraization of mathematics (such as geometry, number theory, and topology) and logic. Various smaller adjustments would be needed for the different topics discussed in the article but they can be addressed later. I was hoping to get some feedback on these ideas and possibly other suggestions. I still have to do some research to work out the details. After that, I would start implementing them one at a time but it will probably take a while to address all the points. Phlsph7 (talk) 14:03, 11 January 2024 (UTC)

This sounds very well, and I am waiting to see the result. However, there is a probelem that you do not mention: section § History.

This section is very detailed on algebra before the 18th century, at a time where algebra did not yet exist. Most of the results presented are not presently considered as belonging to algebra, except if one consider that "algebra" is a synonym of "theory of equations". For example, among many, Diophantus's work is presented as algebra without saying that Diophantine equation belongs presently to number theory.

On the other hand, almost nothing is said about the evolution of algebra during the 19th and 20th centuries, and fundamental results are not even mentioned, such as Gaussian elimination, Hilbert's work on polynomials, McCaulay, who provided the first method fof reducing the solution of polynomial systems to the univariate case, Kummer's work, etc,

I ignore whether there exist reliable sources for the history of algebra during this period (maybe Dieudonné) but an history section must not be empty on this period. D.Lazard (talk) 12:11, 12 January 2024 (UTC)

Thanks, that's a good point! The history section seems to be overly detailed on early developments while having very little on what came afterward. The solution would probably be to summarize a lot of what is currently there and then add more on the evolution of algebra during the 19th and 20th centuries. I haven't had the time to dig into the reliable sources on the history but I hope that a lack of sources won't be a problem here. Phlsph7 (talk) 13:32, 12 January 2024 (UTC)

Definition and etymology

I suggest to add at the end of § Definition and etymology something like the following. I do not add it boldly myself because I am not very happy with the formulation, and a bettter formulation would be easier to source:

The etymology relates algebra to transformation of equations. Indeed, until the 19th century, algebra was essentially the theory of equations; that is, the art of manipulating equations in view of solving them. This explain why some earlier work such as that of Diophantus were considered as belonging to algebra (presently Diophantine equations belong to number theory). During the 19th century, most of the older problems of algebra were solved by the proof of the fundamental theorem of algebra (existence of complex solutions of polynomial equations) and the introduction of Galois theory (characterisation of the equations that are solvable in radicals). So, the scope of algebra has evolved toward what has been described above, and many texts about the history of algebra refer implicitly to a different definition of algebra. D.Lazard (talk) 15:34, 27 January 2024 (UTC)

That's a good idea. I adjusted your text to better fit the sources and to dodge the controversial topic of whether Diophantus as the supposed father of algebra actually engaged in algebra. Phlsph7 (talk) 17:04, 27 January 2024 (UTC)

Semi-protected edit request on 2 February 2024

Under the Linear algebra section the example coefficient matrix has an error on the third column second row, where it is entered as 9 but the system of linear equations above states that the corresponding element should be 7, as that is the coefficient of x3 in the second equation, a bit of a nitpick but a nit none the lesser. Hubblez (talk) 19:54, 2 February 2024 (UTC)

 Done Thanks. Liu1126 (talk) 20:36, 2 February 2024 (UTC)

Semi-protected edit request on 17 February 2024

s/happend/happened/ 78.119.119.111 (talk) 09:20, 17 February 2024 (UTC)

 Done Hyphenation Expert (talk) 09:50, 17 February 2024 (UTC)

Semi-protected edit request on 21 February 2024

Provide a link to "coefficients" under the "Linear Algebra" section. 2nd paragraph, third sentence Techimanz (talk) 16:18, 21 February 2024 (UTC)

 Done
— Urro^{[talk][edits]} ⋮ 16:33, 21 February 2024 (UTC)

Semi-protected edit request on 21 February 2024 (2)

Change "true independent" to "true, independent" under section "Elementary algebra" 3rd paragraph, 3rd sentence Without the comma it appears to indicate truly independent Techimanz (talk) 16:26, 21 February 2024 (UTC)

 Done
— Urro^{[talk][edits]} ⋮ 16:33, 21 February 2024 (UTC)

GA Review

This review is transcluded from Talk:Algebra/GA1. The edit link for this section can be used to add comments to the review.

Reviewer: Bilorv (talk · contribs) 19:41, 9 March 2024 (UTC)

I'm excited to take this one. An enormous topic, obviously, and so I'll spend quite a bit of time thinking hard about scope and breadth (criterion 3). That this article should exist at all is non-obvious and controversial—as the definition and history sections say, there are different interpretations of what "algebra" is. Elementary/symbolic algebra is used in every field of maths and by "algebra" laypeople often refer to notation rather than concept. "Abstract" algebra is a better-defined field of maths but one of the broadest, and one that overlaps with unexpected fields. Nonetheless, I think there is enough of a connection between elementary and abstract algebra that there is an encyclopedia article in it, not just a dictionary definition or a disambiguation page to elementary algebra and abstract algebra. — Bilorv (talk) 19:41, 9 March 2024 (UTC)

Hello Bilorv and thanks for doing this review. These were also some of the concerns I was struggling with when I got started with this article and I arrived at a similar conclusion. Phlsph7 (talk) 07:58, 10 March 2024 (UTC)

Okay, apologies for the verbosity that follows but I'd rather be unambiguous than concise. Feel free to push back on points or make counterproposals. — Bilorv (talk) 23:33, 9 March 2024 (UTC)

Scope comments

• Maybe in "Other branches of mathematics", I feel more could be made of the frequent transformation between geometric problems and algebraic problems. An illustrative example: we can prove that a line intersects a circle either 0, 1 or 2 times by defining some axes and showing that the algebraic equations $y=mx+c$ and $(x-a)^{2}+(y-b)^{2}=r^{2}$ yield a quadratic whose discriminant determines the number of solutions. Algebraic symbol-pushing is a completely different conceptual approach to geometric reasoning. "Geometry" might be its own subsection.

  Good idea, I added a similar example. I considered having separate subsections for the applications of algebra to different mathematical fields but my impression was that this would lead too much into details and is better covered in separate articles, like Algebraic geometry. Phlsph7 (talk) 12:45, 11 March 2024 (UTC)

• Perhaps in "Elementary algebra" or in "History", I feel there should be some treatment of notational conventions. For instance, we typically name an isolated variable x; an equation of a line is typically $y=mx+c$ rather than $A=bC+\theta$ (on a b-A axis graph); arbitrary numbers of variables typically use subscripts like $x_{i}$; polynomials often order terms by decreasing order and coefficients precede variables; vectors might be lowercase bold and matrices uppercase bold. At least noting the existence of conventions with an example would be good.

  I found a spot in the subsection "Elementary algebra" to explain the basics. I left out the part about vectors and matrices since they are not named in any of our examples. Phlsph7 (talk) 13:35, 11 March 2024 (UTC)

• In elementary algebra, some authors make a distinction between "equation" and "identity" and the latter can be an object of study in its own right (like proving that $\cos 3\theta {\text{cosec}}\theta +{\frac {\cos \theta }{\sin 3\theta }}\equiv 2\tan 2\theta$, though I don't know where trigonometry falls with respect to algebra). Identities might deserve some sort of treatment (a possible place where it would naturally follow: the end of the paragraph containing "their truth value usually depends on the values of the variables"). Formulas also, and it seems it is invoked in only one of the three definitions the articles gives of "elementary algebra" (where another word may fit better).

  I added an explanation about the difference between identity equations and conditional equations, which, I assume, is what you meant. I'm not sure if this information is essential but it probably does not hurt either. I didn't get your last comment about formulas. Phlsph7 (talk) 14:02, 11 March 2024 (UTC)

  Yes, that addresses identities/conditional equations. On "formula": the article mentions the word twice ("examines how formulas may be transformed" and "making it possible to express equations as mathematical formulas") but doesn't define it. It might be better just to avoid using this word. To some authors "formula" has a specific definition (e.g. 'an equation in two or more variables') that may not agree with other specific definitions. — Bilorv (talk) 21:35, 16 March 2024 (UTC)

  Ah, I see. I manage to reformulated those passages. Phlsph7 (talk) 08:57, 17 March 2024 (UTC)

• Is abstract algebra only about binary operations? Am I right in saying a normed vector space has two binary operations ($+,\times$) and a 1-ary operation (the norm)?

  According to the reliable sources, the answer is yes to both questions, which kind of creates a paradox. I added a short footnote to clarify this point. Phlsph7 (talk) 16:45, 11 March 2024 (UTC)

• Perhaps in "Definition and etymology", can we talk about how mathematical ideas might be categorised today by whether they are "algebra" or not? e.g. an exam syllabus; Dewey Decimal 512; Mathematics Subject Classification doesn't separate "algebra" as its own thing.

  In principle yes, but I'm not sure that there is an easy answer to this. For the Mathematics Subject Classification, the answer would be quite messy and probably not particularly useful to the reader. Does the Dewey Decimal Classification provide a precise definition of each of its branches or does it rely on the people who use it to classify items correctly? Phlsph7 (talk) 16:54, 11 March 2024 (UTC)

  From my experience the classification just gives the name of an area, like "Algebra", and it is down to each library to sort books into those areas as it wishes. In practice, however, there are databases with book-classification pairs that many libraries use. I think the Dewey number and maybe Library of Congress QA 150-272.5 are significant to mention as they imply that most libraries sort books by whether or not they are about "Algebra". — Bilorv (talk) 21:35, 16 March 2024 (UTC)

  I added a footnote to cover the different classification systems. Phlsph7 (talk) 09:29, 17 March 2024 (UTC)

• In "Elementary algebra", substitution of variables by algebraic expressions is treated as a method of solving, but variables can be substituted for numbers and substitution can be something to study in its own right e.g. substituting natural numbers into $an+b$ generates an arithmetic progression.

  I mentioned substition for numbers. I don't think that the arithmetic progression is central enough to elementary algebra to deserve a separate discussion. Phlsph7 (talk) 17:00, 11 March 2024 (UTC)

• "Elementary algebra has applications in many branches of mathematics, the sciences, business, and everyday life" and "Because of its presence throughout mathematics, the influence of algebra extends to many sciences and related fields" – To spell out every 'application' of 'algebra' would be a neverending task, but I feel that "In various fields" could be renamed "Applications" with an introductory section or a subsection beginning with these ideas. It might say that algebraic notation and terminology is used in science (e.g. chemical equation with its own idea of "balancing"; to express physical laws like $F=ma$), in computer science (with its own idea of variables), and anywhere it is desirable to model systems that change over time (business/economics/geography) or have systems of calculations (finance).

  Done. This topic is a little difficult to properly cover since most sources only talk about specific applications rather than the general overall influence. Phlsph7 (talk) 17:57, 11 March 2024 (UTC)

• I think isomorphism needs some treatment in "Universal algebra", the key being: it is a type of homomorphism that preserves all structures, so in some sense two isomorphic objects are "the same", and we say things like "there are 3 groups of order 8 (up to isomorphism)". Perhaps also automorphism might be worth describing (as it feels slightly different—rather than saying "two structures are the same" it says "one structure has symmetry" or "these elements of the structure have the same role").

  I added an explanation of isomorphisms. You are probably right to say that this should be discussed. I hope it doesn't add too much to the difficulty of this section. Phlsph7 (talk) 18:38, 11 March 2024 (UTC)

• In "Education", can there be some treatment of the psychology of algebra or the reason it is deferred until secondary education? For instance, it might be that algebra is too abstract until a certain maturity or prerequisite knowledge/skills—I don't know if there's some model like concrete-pictorial-abstract that could explain this. Or: is it only notation that is deferred to secondary education (do students discover commutativity when learning times tables by rote, or is it even taught to them explicitly?)?

  I expanded the introductory explanation of that point. The details of how it is taught probably differ a lot form country to country, both concerning the specific curriculum and at which grade primary education ends and secondary education starts. Generally speaking, I would assume that primary students already train some forms of algebraic thinking when solving certain word problems but the topic is not explained to them in a systematic manner. Phlsph7 (talk) 09:20, 12 March 2024 (UTC)

Prose comments

Lead

I have a lot to say about the first paragraph as it defines the scope of this article and its existence as a coherent topic that contains subtopics like elementary algebra and abstract algebra.

• I'm not sure how formal it is to use phrases like "algebra is interested in", "algebra studies" etc. as if it is sentient. Alternatives would be that algebra "encompasses", "includes", "comprises" etc. This is a pattern from top to bottom e.g. "It [education] aims to familiarize students" could be "Educators aim to familiarize students".

  I understand your concern but it is often the most convenient way to express something and should be easily understandable to readers. This language is also used in reliable sources. For example, from [6]: An important part of algebra, linear algebra, studies linear spaces,...; and from [7]: Algebra ... seeks to solve equations ... abstract algebra is interested in such question as.... I wouldn't recommend a general change to this practice but I'm open to fixing individual cases where this type of language does not fit well. Phlsph7 (talk) 09:32, 12 March 2024 (UTC)

• First sentence: I think self-reference ("Algebra ... studies algebraic systems") is to be avoided and I personally don't like the focus on "systems" (less clear than something like algebraic structure) and "equations" (inequalities are also of interest, as are algorithms like "simplifying an expression" that go beyond the equivalence relation properties of equations). Would it be fair to say instead:

  Algebra is the branch of mathematics that encompasses abstract mathematical objects and the manipulation of such objects.

  Or: the first sentence doesn't necessarily have to be a definition e.g. "Algebra is a mathematical concept that includes ..."

  The main difficulty I see is to find a definition that covers the different branches of algebra. I think your first suggestions does a good job at that. One difficulty may be that it is too wide since algebra does not study all abstract mathematical objects. For example, proofs can be understood as mathematical objects but they are not directly studied by algebra.

  According to MOS:LEADSENTENCE, If its subject is definable, then the first sentence should give a concise definition. So starting with a definition would definitely be preferable. What do you think about talking of algebraic structures instead of algebraic systems. Algebaric structures can be defined without reference to algebra. The sentence could be: "Algebra is the branch of mathematics that studies algebraic structures and the manipulation of equations within those structures." If the focus on equations is too narrow, we could replace it with "statements" or "algebraic expressions". Phlsph7 (talk) 09:59, 12 March 2024 (UTC)

  I'd be happy enough to see Algebra is the branch of mathematics that studies algebraic structures and the manipulation of statements within those structures. I see what you mean about my first suggestion being too broad. — Bilorv (talk) 21:35, 16 March 2024 (UTC)

  Done. Phlsph7 (talk) 09:32, 17 March 2024 (UTC)

• I don't like "regular numbers" at all (especially as "regular" is often used for maths jargon). How about: "It is a generalization of arithmetic that introduces variables and algebraic operations other than ..."?

  Done. Phlsph7 (talk) 10:02, 12 March 2024 (UTC)

• Is it possible to say a bit more on the abstract algebra side about how numbers themselves are generalized? In algebraic structures, we treat all sorts of things as though they are numbers (by taking a set of them and applying operations to them): the congruence class 3 (mod 7); 4x6 matrices; symmetries of a hexagon; or just formal variables. Is that abstraction part of what defines algebra?

  We could provide some examples but I usually try to keep the lead as concise as possible so it may not be the right place for this. Phlsph7 (talk) 12:26, 12 March 2024 (UTC)

• I think [[equation solving|isolate variables]] is an Easter egg/overly specific link that could just be dropped e.g. b is isolated in the inequality $b\leq -3$ or the equation $b=cx+d$ (but is changing the subject of a formula the same as "solving"?).

  Done. Phlsph7 (talk) 12:27, 12 March 2024 (UTC)

• "investigates more abstract patterns that characterize algebraic structures" – is it worth saying (or even accurate to say!) universal algebra is about studying classes of algebraic structures?

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

• To stick to the recommended four paragraphs of lead, can "Algebra is relevant to ..." be merged with the previous paragraph?

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

Up to "Abstract algebra"

• Not all variables are maths formatted, like "x-y-pairs" without italics, and the minus sign is its own thing, which needs fixing in phrases like "either 2 or -2".

  Done. I hope I got everything. Phlsph7 (talk) 12:49, 12 March 2024 (UTC)

• In note (a), "an algebraic operation is mapping" should be "is a mapping", but would "function" be better than "mapping"?

  Done. Phlsph7 (talk) 12:49, 12 March 2024 (UTC)

• For consistency I think "polynomial" should be defined in a footnote.

  Done. Phlsph7 (talk) 13:25, 12 March 2024 (UTC)

• "A set is a collection of elements" – I think "unordered" and "distinct elements" needs to be said somewhere in the definition of "set".

  Done. Phlsph7 (talk) 13:25, 12 March 2024 (UTC)

• It's not clear from the prose that The Compendious Book... is alternately titled Al-Jabr.

  I reformulated it to clarify this point. Phlsph7 (talk) 13:25, 12 March 2024 (UTC)

• "it is possible to express a general law that applies to any possible combinations of numbers" – "commutativity" should be mentioned and linked as the example used here.

  Done. Phlsph7 (talk) 13:25, 12 March 2024 (UTC)

• "Elementary algebra is interested in algebraic expressions, which are formed" – As well as the "interested in" issue from above, I feel this would be more concise and direct as "Algebraic expressions are formed ..."

  Done. Phlsph7 (talk) 13:25, 12 March 2024 (UTC)

• Is $\neq$ an important enough inequation to be mentioned?

  Done. Phlsph7 (talk) 13:25, 12 March 2024 (UTC)

• "To achieve this, it relies on different techniques used to transform and manipulate statements" – This would be simpler as "Techniques to transform and manipulate statements are used to achieve this"

  Done. Phlsph7 (talk) 13:47, 12 March 2024 (UTC)

• "A key principle guiding this process is that whatever is done to one side of an equation also needs to be done to the other side of the equation." – Again I'm not sure whether this needs limiting to equations as this is also broadly what is done with inequations too (though multiplying and dividing by non-positives can break things). "whatever is done" might also more formally be "whatever (algebraic) operation is applied" (you can't "do" things like "append a '3' digit to the end of each number" and keep balance).

  I implemented the second part. For the sake of simplicity, I think it's better to just stick with "equation" here. Phlsph7 (talk) 13:47, 12 March 2024 (UTC)

• On isolating variables and $x-7=4$: a key idea is the inverse of an operation or the inverse of an order of steps (to isolate x in $3x^{2}+2=c$, note that x has had $^{2},\div 5,+2$ done to it so we perform $-2,\times 5,{\sqrt {}}$).

  That's a good point. It's just that we haven't defined the inverse of an operation so it might be better to just have the simple intuitive example here and leave the details of the different approaches to the article Equation solving. Phlsph7 (talk) 13:47, 12 March 2024 (UTC)

• "The polynomial as a whole is zero if one of its factors is zero" – This is an "if and only if" (otherwise we wouldn't have solved completely, just found a subset of solutions).

  Done. Phlsph7 (talk) 13:47, 12 March 2024 (UTC)

• "Other techniques rely on commutative, distributive, and associative properties" – These terms and links might best be deferred to "Abstract algebra". I'm not sure what these techniques might be other than those already mentioned plus expanding (which might be worth mentioning).

  Done. Phlsph7 (talk) 13:47, 12 March 2024 (UTC)

• "For example, the system of equations ..."– These might be better as bullet points or (a), (b), (c) than numbers, to avoid confusion. Also, is it not more common to show matrix multiplication without a symbol, and the dot might be confused with the dot product.

  Done. Phlsph7 (talk) 17:53, 12 March 2024 (UTC)

• Rather than a footnote, I think multiplying an equation by a constant is an important enough technique to say (and source) in prose.

  Done. Phlsph7 (talk) 17:53, 12 March 2024 (UTC)

• "Addition is its binary operation and takes two numbers as input to produce one number in the form of the sum as output" – I think this wordiness will create more confusion than it solves. I would prefer "... has the natural numbers as the underlying set and addition as its binary operation". You could add to the "black box" image caption: "... as output, like addition and multiplication do".

  Done. Phlsph7 (talk) 17:53, 12 March 2024 (UTC)

• "The underlying set can contain mathematical objects other than numbers and the operations are not restricted to regular arithmetic operations" – I think showing rather than explaining would be more helpful. A symmetry group might make a good example.

  Done. Phlsph7 (talk) 17:53, 12 March 2024 (UTC)

• The ring of integers is a ring of the form $\langle \mathbb {Z} ,+,\times \rangle$. – I don't think "of the form" fits because there is only one ring of integers and what's given is just notation for it. Maybe "a ring denoted by"?

  Done. Phlsph7 (talk) 17:53, 12 March 2024 (UTC)

• "The rational numbers, the real numbers, and the complex numbers each form a field" – since operations are emphasised here, "... form a field with the operations addition and multiplication"?

  Done. Phlsph7 (talk) 17:53, 12 March 2024 (UTC)

From "Universal algebra"

• "It is a generalization of identity in the sense ..." – I think this sentence is redundant to the previous sentence (and the meaning of the prefix "quasi-").

  You are right. My reason for adding it anyways was that it is a difficult topic of which the reader may not have heard before. So having some redundancy could be a good thing. Let me know if you think otherwise then I'll remove it. Phlsph7 (talk) 18:35, 12 March 2024 (UTC)

• "A homomorphism is a function that... Its special feature is that..." – Can the wordiness be improved? Most of the key information is conveyed in the following sentences. I would just replace these two sentences with "A homomorphism is a function from one set to another that preserves some of the algebraic structure".

  I reformulated this passage to make it more concise. Phlsph7 (talk) 18:35, 12 March 2024 (UTC)

• "in the form of a theory of equations" – I think this phrase needs explaining, at the very least in a footnote.

  I reformulated the sentence. Phlsph7 (talk) 18:35, 12 March 2024 (UTC)

• In the caption, should "al-Khwarizmi's ..." be lowercase even at the start of a sentence?

  No. I fixed it. Phlsph7 (talk) 18:35, 12 March 2024 (UTC)

• "the use of zero and negative numbers" – is a link to 0 justified (as a way to learn more about its history)?

  Done. Phlsph7 (talk) 18:35, 12 March 2024 (UTC)

• vector algebra is a disambiguation link and also has unclear meaning (is this the invention of the Euclidean vector and associated notation and basic operations?)

  I disambiguated it to Vector space. The basic idea here is that operations on vectors were conceived in terms of algebraic structures. Phlsph7 (talk) 18:35, 12 March 2024 (UTC)

• "The basic idea of the even more general" – I would drop "basic", and "first conceived" in the same sentence should just be "conceived".

  Done. Phlsph7 (talk) 18:35, 12 March 2024 (UTC)

• "Closely related developments" – were these all 20th-century developments?

  Most of them but I think not all. The formulation "Closely related developments" is intentionally vague to not imply too much. Phlsph7 (talk) 18:35, 12 March 2024 (UTC)

• "Another key aspect is to apply structures to model how different types of objects interact" – I'm not sure I understand the meaning of this sentence.

  I simplified this sentence. Phlsph7 (talk) 09:00, 13 March 2024 (UTC)

• the connective "if...then" – I think the spacing of MOS:ELLIPSIS should apply here.

  Done. Phlsph7 (talk) 09:00, 13 March 2024 (UTC)

• "students need to learn how to transform them according to certain laws until the unknown quantity can be determined" – The "until" clause isn't always the aim (e.g. "write $y={\frac {2}{5}}x+3$ in the form $ax+by+c=0$").

  I reformulated the sentence to not imply that this is the only goal. Phlsph7 (talk) 09:00, 13 March 2024 (UTC)

• "A common example to introduce students" – Rather than an "example" I feel like this should have some technical name like "mental model" or "pictorial approach".

  Done. Phlsph7 (talk) 09:00, 13 March 2024 (UTC)

• "The mass of some weights" – Maybe this is just me but the mixture of "mass" and "weight" is confusing. A weighing scale moves according to weight, right, so I'd say "the weight of some objects"?

  True, that can be confusing. I kept the term mass since the examples usually focus on the mass of objects rather than the forces acting on those objects. Phlsph7 (talk) 09:00, 13 March 2024 (UTC)

• "For example, students may be presented with a situation in which Naomi" – I think the solution should be given, in prose or a footnote: $2x+x=12$, $x=4$ and Naomi has $2x=8$ apples.

  Done. I slightly modified the example so that x represents Naomi's apples. Phlsph7 (talk) 09:00, 13 March 2024 (UTC)

Referencing and other

Most of the references are introductory books on math/algebra, clearly by expert authors and reliable publishers. They go beyond Wikipedia's requirements of mere verifiability into accessibility to layperson readers or interested learners. Math encyclopediae and books for laypeople like the Very Short Introduction series are suitable here as we only want an overview of an enormously broad topic. Britannica is a source to assess case-by-case and for the (elementary) facts it verifies I think it's a good reference. I've read a bit more into MathWorld and the Stanford Encyclopedia of Philosophy and am happy they are reliable. Other sources are clearly academic and appropriate.

I've been spotchecking ad hoc as I go so I'm only going to do a few systematically (chosen by random number generator): #22, #41, #44, #87, #95. No issues found. Also obviously no copyright issues. Very impressive.

The images are all free and strike a good balance of illustrating concepts and providing historical information. I'm not 100% sold that File:Venn A subset B.svg illustrates much of use (to me the important idea is that operations can't take you out of the subset) but I understand its relevance; however, I think it should be moved a paragraph down so it precedes the subalgebra paragraph as the use of A and B and existing placement could make a reader think it is about the homomorphism example. (Remember that most readers are on mobile and see an image directly where it is placed in wikitext.) — Bilorv (talk) 21:35, 16 March 2024 (UTC)

Done, I moved the image. Phlsph7 (talk) 09:40, 17 March 2024 (UTC)

Overall

Thanks for everything you've done so far and I'm looking forwards to discussing and reviewing this further! From what I've briefly spotchecked so far, it looks like the references are all very reliable, accessible and direct but I'll have to check verifiability at the next stage. This is formally on hold but that might be for longer than seven days (especially given the scope of the topic and that some of my comments might require significant amounts of research), as long as progress is being made. — Bilorv (talk) 23:33, 9 March 2024 (UTC)

@Bilorv: Thanks for the in-depth review and all the helpful and actionable comments. I tried to address all the points and look forward to hearing your responses. Phlsph7 (talk) 09:12, 13 March 2024 (UTC)

I've been broadly following the changes as you've made them but a proper response may have to wait until the end of the week. Things are looking good so I'll move onto spotchecks and a second runthrough for any minor tweaks that need to be made. — Bilorv (talk) 16:50, 13 March 2024 (UTC)

@Phlsph7: I'm happy that almost all of these have been addressed, with a reply on the points about formulas, the Dewey Decimal Classification and the first sentence. None of these are dealbreakers if you prefer the status quo. I've also added a subsection commenting on references and one point on images. — Bilorv (talk) 21:35, 16 March 2024 (UTC)

@Bilorv: I've made the corresponding adjustments. I appreciate all the time and effort you have put into this review! Phlsph7 (talk) 09:42, 17 March 2024 (UTC)

Pass for GA. Thanks for all your work in this review, too. It's an extraordinary achievement to get this vital article to such a high standard. — Bilorv (talk) 09:56, 17 March 2024 (UTC)

Did you know nomination

The following is an archived discussion of the DYK nomination of the article below. Please do not modify this page. Subsequent comments should be made on the appropriate discussion page (such as this nomination's talk page, the article's talk page or Wikipedia talk:Did you know), unless there is consensus to re-open the discussion at this page. No further edits should be made to this page.

The result was: promoted by Hilst talk 13:35, 2 April 2024 (UTC)

(

• Comment or view
• Article history

)

• ... that elementary algebra generalizes arithmetic while abstract algebra generalizes elementary algebra? Source:
  • ALT1: ... that algebraic methods were already studied in the ancient period even though an abstract symbolic notation was not developed until the 16th and 17th centuries? Source:
  • ALT2: ... that the word algebra originates from an Arabic term for the surgical treatment of bonesetting? Source:
  • Reviewed: Template:Did you know nominations/Alfred de Laage de Meux
  • Comment:

Citations

References

Sources

• Maddocks, J. R. (2008). "Algebra". In Lerner, Brenda Wilmoth; Lerner, K. Lee (eds.). The Gale Encyclopedia of Science (4th ed.). Thompson Gale. ISBN 978-1-4144-2877-2. Archived from the original on 2024-01-12. Retrieved 2024-01-13.
• Pratt, Vaughan (2022). "Algebra". The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. Archived from the original on 29 January 2024. Retrieved 11 January 2024.
• Wagner, Sigrid; Kieran, Carolyn (2018). Research Issues in the Learning and Teaching of Algebra: The Research Agenda for Mathematics Education, Volume 4. Routledge. ISBN 978-1-135-43421-2. Archived from the original on 2024-01-12. Retrieved 2024-01-13.
• Tanton, James (2005). Encyclopedia of Mathematics. Facts On File. ISBN 978-0-8160-5124-3.
• Kvasz, L. (2006). "The History of Algebra and the Development of the Form of Its Language". Philosophia Mathematica. 14 (3). doi:10.1093/philmat/nkj017.
• Merzlyakov, Yu. I.; Shirshov, A. I. (2020). "Algebra(2)". Encyclopedia of Mathematics. Springer. Archived from the original on 7 April 2023. Retrieved 11 January 2023.
• Corry, Leo (2024). "Algebra". Encyclopædia Britannica. Archived from the original on 19 January 2024. Retrieved 25 January 2024.
• Cresswell, Julia (2010). Oxford Dictionary of Word Origins. OUP Oxford. ISBN 978-0-19-954793-7. Archived from the original on 2024-01-27. Retrieved 2024-01-27.
• OUP Staff. "Algebra". Lexico. Oxford University Press. Archived from the original on 2013-11-20.

Improved to Good Article status by Phlsph7 (talk). Self-nominated at 12:39, 17 March 2024 (UTC).

Number of QPQs required: 1. Nominator has 18 past nominations.

Post-promotion hook changes will be logged on the talk page; consider watching the nomination until the hook appears on the Main Page.

General: Article is new enough and long enough
New enough: Long enough:

Policy: Article is sourced, neutral, and free of copyright problems
Adequate sourcing: Neutral: Free of copyright violations, plagiarism, and close paraphrasing:

Hook: Hook has been verified by provided inline citation
Cited: Interesting:

QPQ: Done.

Overall: Looks good! Article was nominated within 7 days of achieving Good Article status. Article is over 1500 words of prose. I found no problems with sourcing. Earwig picked up an unlikely violation of 23.1%. AGF on print sources. ALT2 seems to be the most accessible to people since ALT0 and ALT1 rely on technical knowledge in algebra. Nominator only has 18 nominations and does not need a 2nd QPQ at this time (20 nominations required). lullabying (talk) 05:14, 25 March 2024 (UTC)

Stop misappropriating facts and history

Stop misappropriating facts and history. The Arabs did nothing more than translate and preserve texts and did not add a line to the body of already known mathematical theories.  Basic roots of Algebra can be traced to Diophantus and Theon and Algebra is derived from the word Algorithms, it is so obvious.  Plus the alphabet is used for the equations and not the Arabic script.

Now, after the acceptance of the finding of the two Greek mathematicians and the publication, after a crisis, in one of the strictest journals in the field, in the Japanese SCIAMVS (14, 2013 41-57), we can say that now they will probably be traced to different direction the basic roots of Algebra. Diophantus and Theon point in this direction. 2A02:85F:F80C:D100:9C0:4C78:DDF4:7AC1 (talk) 11:54, 8 June 2024 (UTC)

Umm... can you clarify what does that means? I cannot comprehend your words. Dedhert.Jr (talk) 13:27, 8 June 2024 (UTC)

nothing more than translate and preserve texts – this is directly contradicted by your cited source, which (in the one relevant half-sentence) describes the flourishing of algebra in the Islamic world. –jacobolus (t) 23:20, 8 June 2024 (UTC)

The IP seems to suggest that the Arabs did not make any significant contribution to algebra. This goes against what the sources in our article say and, as jacobolus has pointed out, is not even supported by the source they refer to. Phlsph7 (talk) 07:51, 9 June 2024 (UTC)

Agreed, it is very well-known that Arabic contributions to algebra were significant. Mathwriter2718 (talk) 20:42, 25 July 2024 (UTC)

For what it's worth, there are some other point of confusion here. The word algorithm comes from the name Al-Khwārizmī. The word algebra comes from Al-Khwārizmī's book Al-Jabr (al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah) which was translated into Latin as Liber Algebrae et Almucabola. What we call "algebra" today, i.e. pushing alphabetic symbols around using mathematical notation, is formally quite different than the kind of algebra from Theon or Al-Khwārizmī. The solution steps may be conceptually the same, but modern conventions for labeling variable quantities, writing equations, etc. developed starting in the 16th century. –jacobolus (t) 17:22, 9 June 2024 (UTC)

Somma di arithmetica

@Bjørn Leon Søren Riedel Thank you for your contribution, but I've undone your edit for two reasons. One is that this is word-for-word from the source. Even if the source is public domain, you should still either summarize what it says in your own words, or present it as a direct quote with attribution. The other problem is that claims of something being "the first" or "the earliest" are hard to be sure of and often turn out to be incorrect. I note that Christies, when they sold a copy of this book, said "is often regarded as the first printed book on algebra", which is a much more justifiable statement than unequivocally calling it "the earliest". RoySmith (talk) 15:54, 8 August 2024 (UTC)

Another potential problem is that the source for this claim is almost 150 years old. Maybe this view was accepted back then but contemporary scholarship presents a very different picture. Phlsph7 (talk) 16:01, 8 August 2024 (UTC)

Incompleteness of the article

As I noted in Wikipedia:Featured article candidates/Algebra/archive1#D.Lazard, this article is very imcomplete. It is also misleading by giving a undue weigth to universal algebra, which is a minor branch that is almost never used outside itself and does not contain any notable theorem. On the other hand, most major branches of algebra are unduely mentioned in passing only, without any mention of their major achievements.

Fixing this major issue will require a lot of work. For starting this, I'll add new sections for some of the major branches of algebra, with {{main article}} templates, {{expand section}} tags, and sometimes a few worda on their main achievements. D.Lazard (talk) 15:01, 27 September 2024 (UTC)

Thanks for the suggestions. Several of the links you mentioned are already discussed in the article, such as the Abel–Ruffini theorem and the Fundamental theorem of algebra, which are covered in the history section. Phlsph7 (talk) 15:33, 27 September 2024 (UTC)

I'm not sure that adding these new subsections as additional branches is the right way to go. Polynomials are currently already discussed in the subsection "Linear algebra". Group theory is discussed in the subsection "Abstract algebra". Phlsph7 (talk) 15:35, 27 September 2024 (UTC)

It might be more effective to find a source(s) that describes which branches of algebra are the main ones, and work from there. The current addition of new empty sections doesn't really seem to move this article forward Aza24 (talk) 22:00, 27 September 2024 (UTC)

I broadly agree with D.Lazard that the current article does not really fairly represent the breadth of "algebra". (I would not personally consider it to be close to a "good article" by Wikipedia standards, just based on current scope.) Adding stub sections seems fine, as a target to be fleshed out, and indeed there could probably be more such sections. Some existing sections should perhaps be shortened. An incomplete list of topics off the top of my head which are not currently mentioned, or barely mentioned, but should perhaps be discussed more seriously and put into better context, include complex number, multilinear map, tensor, linear form, dual space, quaternion, exterior algebra, Clifford algebra/geometric algebra, Diophantine equation, field extension, root of unity and cyclotomic polynomial, lattice (order), lattice (group), discrete group, finite field, rational function, all of the objects of algebraic geometry and algebraic topology, Gröbner basis, Lie group, Lie algebra, symmetry group, equivalence relation and equivalence class, representation theory, category theory, linear subspace, Gaussian elimination, eigenvalues and eigenvectors, matrix decomposition (esp. the singular value decomposition), polynomial ring, Hilbert space, spectral theory, discrete Fourier transform, .... –jacobolus (t) 23:25, 27 September 2024 (UTC)

My impression is that these scope-disagreements arises from what we understand with the term "algebra". Some mathematicians use the term to refer primarily to abstract or modern algebra in contrast to elementary or classical algebra. This would exclude most of the algebra taught in school and only focus on advanced topics encountered in undergraduate and graduate studies. If this should be the scope of the article then, I agree, it is too brief about many of the advanced topics. However, this article is written for the general public, including high school students taking basic algebra classes, not just for mathematicians. This is why I think the article should try to cover both the basic and the advanced algebra. In this case, we don't have the space to go into all the advanced details and have to be picky about which of the more advanced topics we explain.

One difficulty with many sources for this subject is that they only focus on one meaning of "algebra", such as the books D.Lazard mentioned here, which are called "Algebra" but focus on advanced algebra. For example, Lang's book corresponds to a "one-year course in algebra, at the graduate level" (Lang 2002 p. v) and presupposes knowledge of "vector spaces, linear maps, matrices" and polynomials (Lang 2002 p. xii). A better source for the purpose of scope might be Pratt 2022, which gives an overview encompassing both the basic and the advanced.

Concerning the added stub sections: many of these topics are already discussed in the article, so adding them again as new sections would cause a lot of overlap. For example, the section "Abstract algebra" already discusses groups. Group theory is usually understood as part of abstract algebra so adding a section "Group theory" is one more branch in addition to the section "Abstract algebra" is problematic. For now, I turned the newly added links in the recent expansion suggestions into see-also links until we figure out how to proceed. Phlsph7 (talk) 07:49, 28 September 2024 (UTC)

I understand this as asserting that the scope of our article "algebra" must exclude what you call "avanced algebra". I strongly disagree: following the general rules of Wikipedia, the scope of the article is everything that is commonly called algebra. The whole content of Lang's book is algebra, but this does not mean that algebra is restricted to the content of this book. In particular, everybody considers that elementary algebra and universal algebra belong to algebra. Also, Pratt's 2022 is a source that must be considered with care, not only because it is a WP:tertiary source: Section "Application" does not mention any application outside universal algebra; section "Algebraic geometry" does not mention commutative algebra; section "Algebraic topology" does not mention homological algebra.

For example, the section "Abstract algebra" already discusses groups. No, it contains the definition of groups. But it does not discuss the subject. Discussing the subject should consists in explaining why groups are important, in saying that groups are used in almost all mathematical areas as well as in Rubik's cube theory, crystallography, quantum mechanics, and many other applied subjects, and mentioning the most important theorems of group theory. This requires a specific section that can and must be written in non-technical terms. D.Lazard (talk) 10:59, 28 September 2024 (UTC)

I don't claim that our article must exclude advanced algebra and I join you in disagreeing with this claim. The article already covers various aspects of it and I'm not opposed to some form of expansion on the advanced math side. We can't go too much into depth and have to leave the details to child articles on more narrow topics. Rather than flood the article with new and overlapping branch-sections, I would advocate a more subtle approach. Many of the applications of group theory are already discussed in the subsection "Applications#Other branches of mathematics", including the Rubik's Cube group you mentioned. I added some of the additional applications you suggested and also took a small step in the direction of mentioning theorems of group theory. This is probably not exactly what you had in mind but I hope it is a step in the direction you intended. I'll see if I can find ways to work more of the suggestions into the text. Phlsph7 (talk) 16:25, 28 September 2024 (UTC)

We can't go too much into depth and have to leave the details to child articles: I agree. But even limited to the most important information, the main branches of algebra cannot be summarized in one or two lines in a section ("Abstract algebra" where nobody will search it.

By the way, I had look on the section § Linear algebra; it must be completely rewritten: it is written in a textbook style, explaining in details what solving means and how to solve a single linear equation. On the other hand it lacks essential information, such that the existence of methods for solving completely any linear system, that the algorithms implementing these methods allow solving routinely systems of more than 100,000 equations with 100,000 unknowns, and that this is used dayly for weather forecasting. This is only an example of lacking information, which could be also interesting to readers that are not mathematicians. There is much other lacking information. In particular several of the items mentioned above by Jacobolus must be mentioned in this section, with comments explaining the relations between them and with linear algebra, and links to the relevant articles. D.Lazard (talk) 17:38, 28 September 2024 (UTC)

I don't think we should assume that everyone knows what exactly it means to solve a system of linear equations. If this article was intended for a math encyclopedia rather than a general encyclopedia, I would agree with your suggestions to reduce the basics and expand on the advanced topics. However, if you look at the FA reviews, there is consensus that making this difficult topic accessible to the general reader is a key strength of the article. Focusing on the advanced aspects of topics, which are of interest primarily to mathematicians, is a problem for many of our math articles. If we followed the road of reducing basics and expanding on advanced topics, we may end up making the article worse, not better.

I'm not sure what you mean by "lacks essential information, such that the existence of methods for solving completely any linear system". As far as I'm aware, the methods of Gauss–Jordan elimination and LU decomposition, mentioned in the paragraph starting with "Many techniques employed...", can be used. Depending on the specific problem, they may not be the most computationally efficient ones. If you have a source in mind that discusses this specific example of solving linear systems in weather forecasting, I can try to include some information on it. Phlsph7 (talk) 08:17, 29 September 2024 (UTC)

Personally I think "Applications#Other branches of mathematics" is a substantially poor organization scheme, and we really need very significant expansion in the "what is algebra" part of the article, which in my opinion should not be rigidly separated from "how is algebra applied". Algebra (of various types) is fundamental to essentially all areas of mathematics (not to mention other technical disciplines), and there are huge portions of mathematics which would be classified as directly "part" of algebra but which are currently unmentioned or barely mentioned on this page. I would recommend eliminating an explicit applications section, and dramatically expand the portions of the article talking about different types of algebra, including enough context to explain what they are actually about, not just nominally mentioning them in passing. –jacobolus (t) 21:13, 28 September 2024 (UTC)

There are different ways to organize algebra topics into sections. Talking about applications in a separate section is one way and is also employed by reliable sources, such as Pratt 2022 and Terras 2018. But it's possible that your suggestion would work as well. I'll try to include some of your proposed expansions in the article but I'm not sure that a dramatic expansion is the right way to go. Phlsph7 (talk) 08:24, 29 September 2024 (UTC)

As explained above, Pratt 2022 is not reliable for anything. Terra 2018 is a textbook, and, per WP:NOTTEXTBOOK, it cannot be used as an example for the organization of a Wikipedia article. D.Lazard (talk) 08:50, 29 September 2024 (UTC)

As far as I can tell, Pratt 2022 fulfills the requirements of reliable sourcing. For example, it's published by Stanford University and peer-reviewed.

As I understand it, WP:NOTTEXTBOOK talks about how Wikipedia articles are written, not about sources used in Wikipedia articles. Having a section called "Applications" is not something that only textbooks do. Phlsph7 (talk) 15:49, 29 September 2024 (UTC)

The Stanford Encyclopedia of Philosophy doesn't seem like a necessarily authoritative source about how to organize the topic of "Algebra". The target audience is philosophers and the target scope is the subtopics of particular interest to philosophy. –jacobolus (t) 18:23, 29 September 2024 (UTC)

Philosophy of mathematics aims to give general characterizations of mathematics. It's not just about collecting mathematical facts that are of particular interest to philosophy. Pratt 2022 is obviously not the only source to consider but it is one high-quality source to consider. Phlsph7 (talk) 07:40, 30 September 2024 (UTC)

I have started to complete the article. For the moment I have written only the (subsub)section devoted to polynomial equations. IMO, this is an example of the style that must be used in this article. This shows also that problems that occupied algebraists during centuries can be summarized if few lines, while remaing comprehensible for a large audience. D.Lazard (talk) 10:46, 29 September 2024 (UTC)

Thanks for making the effort to express this in an accessible way. I'm not sure that we should have this as a separate main branch of algebra since its topics belong to other branches already mentioned. What do you think about turning it into a subsection of the section "Elementary algebra" and have the section "Abstract algebra" cover the part on polynomial rings, which also belongs to ring theory? This is not ideal but it's probably the best fit. Given that this is a wide-scope overivew article, I would also suggest that we do not use any internal subsections and subsubsections for the section "Polynomials" so we just restrict ourselves to a few paragraphs to cover the most important points we can fit in there and leave the rest to child articles. Phlsph7 (talk) 12:05, 29 September 2024 (UTC)

In what I have already written, almost everything is left to linked articles, only the most important points are there. Trying to include them in the previously existing sections would make them confusing, and, overall, make difficult for most readers to find easily what they are looking for (looking on the table of content is much easier than word searching). D.Lazard (talk) 12:58, 29 September 2024 (UTC)

We have sources that the first four subsection of the section "Major branches" are indeed major branches. As far as I'm aware, the term "Polynomials" is usually not understood as a name of a major branch of Algebra in addition to what we already have here. It's not up to us to decide how Algebra is divided into major branches. There may be different ways to address this, but the suggestion in its current form fails to do so. As a sidenote: all the claims within this section also need sources. Phlsph7 (talk) 15:43, 29 September 2024 (UTC)

We have sources that the first four subsection of the section "Major branches" are indeed major branches, but we have other sources that imply exactly the contrary. So, we have to decide what is the most common opinion of mathematicians. Here we have a subject classification elaborated by the whole mathematical community and two major treatises on algebra written by well known mathematicians that are the subject dedicated Wikipedia articles (there are also reliable sources asserting that these books are highly influential; see Serge Lang and Moderne Algebra). None of these three highly reliable sources mention "universal algebra" and "abstract algebra". So, it is clear that for most mathematicians, "abstact algebra" and "universal algebra" are not major branches outside educational community of some countries.

all the claims within this section also need sources: Sure, but this is written in wp:summary style and sources can be found in the main articles and the linked article. Also, WP:LIKELY may be applied. D.Lazard (talk) 17:00, 29 September 2024 (UTC)

If you want to decide what "major branches" are you could pull all of the algebra-related topics from Mathematics Subject Classification. But I'm not convinced that "major branches" is the most useful top-level organization system for this article. –jacobolus (t) 18:19, 29 September 2024 (UTC)

The Mathematics Subject Classification does not explicitly subdivide algebra into branches. Interpreting it ourselves to arrive at a subdivision would involve original research. I agree with Mathwriter2718, who has pointed out that 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. Lang's textbook is about the more advanced topics in algebra. I don't think it provides a general subdivision of algebra, but I would be interested if someone could tell me the relevant page if I'm wrong.

The division of algebra into elementary, linear, and abstract algebra is widely used. For example, see O’Regan 2020, Arcavi et al. 2017, Diedrich & Lovett 2022, and Brešar 2019. It's not so clear how universal algebra fits into that picture. It's sometimes characterized as a branch in addition to abstract algebra but depending on how broad we characterize abstract algebra to be, we could also include universal algebra in it. In this case, we could turn it into a subsubsection of the subsection "Abstract algebra". Phlsph7 (talk) 07:55, 30 September 2024 (UTC)

You could alternately take the "branches of mathematics" section from the Princeton Companion to Mathematics and pull out the ones which are part of algebra. The problem with "elementary / linear / abstract" algebra" is that huge parts of modern mathematics get shoved into on tiny section which cannot be written/organized clearly to make them comprehensible to readers. What you are basically doing with this classification is ignoring the past at least 100 years of work in favor of focusing on what is taught in primary/secondary school, but this gives a misleading impression of the scope of "algebra" as understood in the 21st century. –jacobolus (t) 14:23, 30 September 2024 (UTC)

Here's what the introduction of the Princeton Companion says about Algebra:

2.1 Algebra
The word “algebra,” when it denotes a branch of mathematics, means something more specific than manipulation of symbols and a preference for equalities over inequalities. Algebraists are concerned with number systems, polynomials, and more abstract structures such as groups, fields, vector spaces, and rings (discussed in some detail in some fundamental mathematical definitions [I.3]). Historically, the abstract structures emerged as generalizations from concrete instances. For instance, there are important analogies between the set of all integers and the set of all polynomials with rational (for example) coefficients, which are brought out by the fact that both sets are examples of algebraic structures known as Euclidean domains. If one has a good understanding of Euclidean domains, one can apply this understanding to integers and polynomials.

This highlights a contrast that appears in many branches of mathematics, namely the distinction between general, abstract statements and particular, concrete ones. One algebraist might be thinking about groups, say, in order to understand a particular rather complicated group of symmetries, while another might be interested in the general theory of groups on the grounds that they are a fundamental class of mathematical objects. The development of abstract algebra from its concrete beginnings is discussed in the origins of modern algebra [II.3].

A supreme example of a theorem of the first kind is the insolubility of the quintic [V.21]—the result that there is no formula for the roots of a quintic polynomial in terms of its coefficients. One proves this theorem by analyzing symmetries associated with the roots of a polynomial, and understanding the group that these symmetries form. This concrete example of a group (or rather, class of groups, one for each polynomial) played a very important part in the development of the abstract theory of groups.

As for the second kind of theorem, a good example is the classification of finite simple groups [V.7], which describes the basic building blocks out of which any finite group can be built.

Algebraic structures appear throughout mathematics, and there are many applications of algebra to other areas, such as number theory, geometry, and even mathematical physics.

This is obviously only the barest introduction, but you'll notice that our article here does not even mention the concept of a Euclidean domain, the insolubility of the quintic, or the classification of finite simple groups. Later in the introduction, under "I.3 Some Fundamental Mathematical Definitions", we have "2 Four Important Algebraic Structures" in which several pages are spent on Groups, Fields, Vector Spaces, Rings, and then "3 Creating New Structures Out of Old Ones" which goes another several pages, then "4 Functions between Algebraic Structures" which is another several pages. All of the material in these subsections is directly relevant to this topic and should be covered in enough detail in this article to give readers appropriate context. I don't think it can effectively fit in your current classification. –jacobolus (t) 14:52, 30 September 2024 (UTC)

All of the sources you gave here are focused on education up through about 1st–2nd year undergraduates, and basically sweep everything beyond the high school level into one pile, and then don't really discuss it (specifically, we have one textbook which is a grab-bag of math topics related to computing, with this IMO not-very-useful-to-computing-students chapter on "algebra"; one book for teachers about high-school level algebra; one advice book for undergraduate math majors of which the part you linked is supposed to be a summary of the Mathematics Subject Classification, and the preface of an introductory college algebra book; I understand these are just being invoked as arbitrary examples, but I don't personally think any of these seems like a good source about how to classify the parts of algebra).

As an analogy, this classification would be kind of like an article about "computing" listing as the "branches of computing": "basic arithmetic", "games", and "general-purpose programs", with the latter category including everything else. –jacobolus (t) 14:41, 30 September 2024 (UTC)

Thanks for looking up the text from the Princeton Companion to Mathematics. The topics you mentioned from the text are presented as examples and they could have chosen different ones for their overview. The insolubility of the quintic (Abel–Ruffini theorem) is discussed in our article, as I have pointed out before. Our article also covers various algebraic structures, including Groups, Fields, Vector Spaces, Rings. The part about Creating New Structures Out of Old Ones is covered in our article in the discussions of subalgebra while subrings and field extensions are also mentioned. Functions between Algebraic Structures are addressed by our subsection "Universal algebra" in the discussion of homomorphisms and isomorphisms. So from covering the points you mentioned, I think our article does well. Given the wide scope of the topic and the little space we have, we can only present a Bird's-eye view per Wikipedia:Summary style. But I've heard your concerns and I'll try to include some expansions. The the Princeton Companion to Mathematics could be a helpful source for that, I'll look through the sections you mentioned.

The division into elementary, linear, and abstract algebra is quite common; the sources I gave are only a small sample. Most people only ever get in contact with elementary and linear algebra. Phlsph7 (talk) 16:29, 30 September 2024 (UTC)

"they could have chosen different ones" – yes that's fine, and you're right that we added 1 sentence about degree ≥ 5 polynomials within the past few days (and had a sentence about it in the history section previously). But I guess the point is, I think any article about as broad a mathematical subject as this should be mentioning and putting in context a significant number of theorems from across the breadth of the subject, and we currently are barely scratching the surface. "are addressed by" – in my opinion, they really are not. A reader who goes through this article is not, in my opinion, getting a clear or accurate impression of e.g. what homomorphisms and isomorphisms are or what they are for ("Homomorphisms are tools in universal algebra to examine structural features by comparing two algebraic structures" doesn't seem like a good way of introducing this topic), and as I said previously we have no mention of equivalence classes, and quotient maps. As another example, a one-sentence summary like "Ring theory is the study of rings, exploring concepts such as subrings, quotient rings, polynomial rings, and ideals as well as theorems such as Hilbert's basis theorem." seems completely inadequate to be the extent of the discussion on this article; ring theory should be a whole section, explaining the purpose, context, main results, and relation to other topics.

Most people only ever get in contact with elementary and linear algebra. – That's true, but as a comparable example, most people's direct experience with chemistry is limited to cooking, mixing 2-part adhesives, and using household cleaning products, plus whatever experiments they did in a high school class; I don't think we should limit our article Chemistry to those topics. –jacobolus (t) 19:11, 30 September 2024 (UTC)

I would oppose limiting the chemistry article to most people's direct experience with chemistry while excluding everything else, just as I oppose the same for algebra. This is not what I'm suggesting or what the article is currently doing. Given the breadth of topics, I'm for a Bird's-eye view per Wikipedia:Summary style, not exclusion. Phlsph7 (talk) 08:11, 1 October 2024 (UTC)

Just in case there was a misunderstanding: my suggestion was to have a subsubsection "Polynomials" within the subsection "Elementary algebra". So the term "Polynomials" would be in the table of contents. Phlsph7 (talk) 15:53, 29 September 2024 (UTC)

Your suggestion is nonsensical, because it would imply that the sophisticated methods described in the linked article are elementary algebra. D.Lazard (talk) 17:04, 29 September 2024 (UTC)

I added the corresponding sources and adjusted the content to fit into the subsection "Elementary algebra". Now, most of the main claims are sourced to works explicitly talking about elementary/classical algebra. Phlsph7 (talk) 11:07, 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) 15:43, 30 September 2024 (UTC)

Organization/scope changes I would consider:

• Possibly retitle § Definition and etymology to something indicating a more general summary (e.g. "Introduction", but perhaps there's a better title), and try to give an informal overview of what algebra is like and what it is about. Explicitly contrast algebra vs. arithmetic, algebra vs. geometry, algebra vs. analysis in this section. Include 1–2 paragraphs summarizing the history of the "algebraization"/abstraction of mathematics, giving some idea of different levels of abstraction.
• Eliminate the current § Major branches grouping (this title could perhaps be re-used further down), and put "Elementary algebra" at top level, moving content about applications to § Education into the same section.
• Make a top-level section about the most important and common ("abstract") algebraic objects, including sections about means of constructing new objects from existing ones and different kinds of functions between objects.
• Make a top-level section about the main branches / fields of study in algebra. Substantially reduce the current § Universal algebra section and put it in there (or it could frankly just be cut entirely).
• Either add linear algebra as a separate top-level section, or combine it in the previous section. Pare away most of the content currently in § Linear algebra which is excessively detailed for the context and more or less misses the point, and give a more complete and neutral high-level overview.
• Refocus the § History section, shortening most of the part currently there and dramatically expanding the portion about the past century or two.
• Eliminate or completely rewrite the § Applications section, which would need to be expanded probably 20x to meet WP:NPOV at the depth of discussion of the current content there about § Logic. If there's going to be such a section, recruit some mathematicians to brainstorm a list of applications of algebra.

(As I said above, with current organization and scope, I wouldn't consider this article to be GA by Wikipedia standards, let alone FA.) While we're at it, it might be helpful to open a public call for better sources, especially high-level survey sources. –jacobolus (t) 17:13, 30 September 2024 (UTC)

I essentially agree, and would add a few more points

• Merge sections § Abstract algebra and § Universal algbra into a single section "Algebraic structures". The resulting section must contain something like "the study of algebraic structures in general is often called "abstract algebra" in the educational world". The mention of universal algebra must be reduced to a few lines, and categories must appear here as a common and more powerful substitute to universal algebra (note that homomorphisms and isomorphisms became important in universal algebra only after the introduction of categories that have (homo)morphisms in their basic definition)
• Reduce dramatically the bibliography by removing all popularization articles, all courses that are not regularly published, and most tertiary sources (WP:NOTCATALOG); keep only textbooks with a large audience and many citations, and specialized sources that are needed for some technical points.

D.Lazard (talk) 20:44, 30 September 2024 (UTC)

Thanks for the suggestions. I have problems seeing in what sense they would lead to overall improvements. Generally speaking, I fear the attempt to rewrite this article to fit the Mathematics Subject Classification and target the scope of graduate algebra courses is misguided since Wikipedia is a general encyclopedia, not a math encyclopedia.

More specifically, I'm not a big fan of sections called "Introduction" since this title only indicates the intended purpose of the section but not the content. The current title is much more descriptive. The section already contrasts algebra and arithmetic but I'm not opposed to also mentioning the contrast to geometry and analysis. The division into main branches creates an accessible overview and is well-supported by the sources. The more specific topics can be addressed in this layout without the need for separate main sections for each of the algebraic structures studied in algebra graduate courses. I think the history section is balanced as it is. A dramatic expansion of the last 2 centuries to focus on the more advanced topics while reducing everything else would overemphasize recent discoveries. The earlier developments may seem simple from our point of view but they were important in leading to where we are now and that development is exactly what the history section is about. The applications section has currently about 1000 words. Expanding it 20 times would be 20000 words for a single section. This would be way too long even if our article was exclusively dedicated to applications of algebra, see WP:SIZERULE.

I'm not aware of specific sourcing problems, but they could be addressed one by one. The important point is that everything is supported by a reliable source. As I understand it, WP:NOTCATALOG is about the content of articles, not about the sources to support that content. Phlsph7 (talk) 08:03, 1 October 2024 (UTC)

"the scope of graduate algebra courses is misguided since Wikipedia is a general encyclopedia" – just because Wikipedia is a general encyclopedia doesn't mean that readers aren't interested to know what "algebra" means as a topic in mathematics. "current title [Definition and etymology] is much more descriptive" – yes, it is descriptive of the content currently there, but I don't think the current content makes a very good introductory section and am proposing changing what is written there at which point the title would no longer be appropriate, but what to title this section isn't really important. "division into main branches creates an accessible overview" – no it really does not; it creates an arbitrary, lopsided, and substantially misleading categorization with poor narrative flow and no room to grow. "would overemphasize recent discoveries" – the meaning of "algebra" has radically changed and expanded in the past 2 centuries, and the vast (vast) majority of work in the subject is from the recent past. The current section is like having a "history" section in the article about Cars which spent 90% of its content on chariots, wheelbarrows, and horse-drawn carts, and then mentioned engine-powered automobiles briefly in the last paragraph, justifying that distribution on the grounds that expanding the latter part would "overemphasize recent discoveries". "earlier developments may seem simple from our point of view but they were important" I completely agree. In "summary style" we can give a concise summary of the major developments and leave further details to more specific other articles where the story should be fleshed out in as much detail as Wikipedians have time to write about. "The applications section has currently about 1000 words. Expanding it 20 times would be 20000 words for a single section. – yes, you can see my point then: the current section is completely contrary to NPOV (cf. WP:BALANCE), and must be rewritten (or possibly eliminated and reincorporated in a different organization scheme) because if expanded to be neutral and matching the content already there it would be overwhelmingly large. "important point is that everything is supported by a reliable source" – ideally, especially for a featured article, everything should be supported by the best, most relevant source we can find. –jacobolus (t) 14:16, 1 October 2024 (UTC)