Talk:Algebra - Knowledge

5297:

and rings (discussed in some detail in some fundamental mathematical definitions ). 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 . A supreme example of a theorem of the first kind is the insolubility of the quintic —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 , 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.

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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

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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.

1119:). 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 5313:. 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. – 558:). 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. 4845:

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.

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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. –

1787:"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. 2492:"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). 5328:

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).

364: 532:). 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 354: 333: 300: 2010:. 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". 5122:

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.

445: 5169:, 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 Knowledge articles (there are also reliable sources asserting that these books are highly influential; see 21: 93: 4391: 4852:, 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. 3586: 291: 4386: 949: 875: 799: 721: 222: 141: 4960:, 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. 4941:. 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. 4359: 4351: 4322: 4302: 4291: 4262: 4251: 3472: 4594:, 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. 994: 920: 844: 767: 5253:. 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". 5386:– 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 1029: 2938:"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". 5271:

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

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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

4955:

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,

748:

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

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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

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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

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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

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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

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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 Knowledge, the scope of the article is everything that is commonly called algebra. The whole content of Lang's book is algebra, but

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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

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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

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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

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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

4683:

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 Knowledge standards, just based on current scope.) Adding stub sections seems fine, as a target to be fleshed out,

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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

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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

1998:

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

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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

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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,

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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

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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

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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.

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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 aims to familiarize students" could be "Educators

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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

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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 Knowledge is a

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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

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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

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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. —

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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

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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

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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

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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

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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

617:

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

1687:

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

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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,

524:. 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. 5699:– 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. 5327:

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

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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

1876:

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

2719:"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). 5424:– 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 520:

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

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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".

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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.

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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

1851:

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

1550:

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.

2118:, 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? 4964:

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.

5177:). 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. 1306:

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

5612:(As I said above, with current organization and scope, I wouldn't consider this article to be GA by Knowledge 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. – 1734:

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

4522:, 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. – 1540: 5716:

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.

2738:"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 3518:

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. —

3464:

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

4856:

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.

4933:; 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 3416:

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

5707:– 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 5331:

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.

4647:

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".

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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)

4881:

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

4463:

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.

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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. –

2921:

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.

1436:; 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. 5231:

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

527:

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

4662:

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

4875:: 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. 2780:"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 2679:

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

4632:

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.

2415:"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 ..." 5506:

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.

5571:

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

5725:), 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. 5053:

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.

2803:"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 1302:

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.

5344:

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

5234:. 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. 5477:

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.

5775: 1856:

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").

2473:"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" 1932:

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

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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. —

2869: 5152:

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.

5366:. 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. 4997:. 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. 3255:"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"? 5418:

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.

4926:: 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. 5369:

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.

4993:

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

4420: 2892:"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"? 3425:

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.) —

420: 56: 696:

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.

4570:

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.

3768: 1300: 5800: 3409:

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.

4937:

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

5444:

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

2674: 4957: 4894:, 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. 4849: 4591: 3389:

Most of the references are introductory books on math/algebra, clearly by expert authors and reliable publishers. They go beyond Knowledge's requirements of mere verifiability into

1962:

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

3171: 5674:

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,

4554: 2700:"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). 1689: 1458: 680:). 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. 2635: 5036:

talks about how Knowledge articles are written, not about sources used in Knowledge articles. Having a section called "Applications" is not something that only textbooks do.

3131:"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 1439:

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.

1399: 976:

Change "true independent" to "true, independent" under section "Elementary algebra" 3rd paragraph, 3rd sentence Without the comma it appears to indicate truly independent

5416:"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." 2587: 52: 3215: 2396:"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. 1632: 2176: 1150:

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.

3307: 5790: 2211: 1860:

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.

1364: 1327:

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

1223: 603:

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.

2541: 2452: 1455:

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

1434: 3362: 1818: 1762: 596:

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

5805: 3333: 1067: 600:, 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, 78: 64: 2784:

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".

644:

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:

5755: 3236:"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". 3979: 2232:"investigates more abstract patterns that characterize algebraic structures" – is it worth saying (or even accurate to say!) universal algebra is about studying 5221:

The Mathematics Subject Classification does not explicitly subdivide algebra into branches. Interpreting it ourselves to arrive at a subdivision would involve

2918:"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-"). 1831:

Done. This topic is a little difficult to properly cover since most sources only talk about specific applications rather than the general overall influence.

1057: 4451:– this is directly contradicted by your cited source, which (in the one relevant half-sentence) describes the flourishing of algebra in the Islamic world. – 1771:

I mentioned substition for numbers. I don't think that the arithmetic progression is central enough to elementary algebra to deserve a separate discussion.

4557:, said "is often regarded as the first printed book on algebra", which is a much more justifiable statement than unequivocally calling it "the earliest". 1885:

that is deferred to secondary education (do students discover commutativity when learning times tables by rote, or is it even taught to them explicitly?)?

3504:

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.

3258:

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.

1889:

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.

5815: 1916:

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

410: 5394:– 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 1165:

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. —

5695:– just because Knowledge is a general encyclopedia doesn't mean that readers aren't interested to know what "algebra" means as a topic in mathematics. 5640:

Reduce dramatically the bibliography by removing all popularization articles, all courses that are not regularly published, and most tertiary sources (

3086:"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. 5785: 5703:– no it really does not; it creates an arbitrary, lopsided, and substantially misleading categorization with poor narrative flow and no room to grow. 5029:

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.

3685: 1657:

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.

226: 5795: 2873:– 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"? 5349:(Abel–Ruffini theorem) is discussed in our article, as I have pointed out before. Our article also covers various algebraic structures, including 1034: 1828:), and anywhere it is desirable to model systems that change over time (business/economics/geography) or have systems of calculations (finance). 1946:. 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. 1566:

authors "formula" has a specific definition (e.g. 'an equation in two or more variables') that may not agree with other specific definitions. —

5810: 5557:

1–2 paragraphs summarizing the history of the "algebraization"/abstraction of mathematics, giving some idea of different levels of abstraction.

5310: 386: 207: 5492:

Your suggestion is nonsensical, because it would imply that the sophisticated methods described in the linked article are elementary algebra.

5770: 5582:

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

5564:

grouping (this title could perhaps be re-used further down), and put "Elementary algebra" at top level, moving content about applications to

4597:

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

4149: 4058: 4029: 3973: 3941: 3929: 3901: 3888: 3875: 3862: 3837: 3821: 3808: 2286: 618:

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.

243: 29: 5066:

Philosophy of mathematics aims to give general characterizations of mathematics. It's not just about collecting mathematical facts that are

4848:

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

543:). Several high-quality overview sources I checked present linear algebra and universal algebra as major subfields of algebra (for example, 827: 5586:

which is excessively detailed for the context and more or less misses the point, and give a more complete and neutral high-level overview.

4035: 3274:"For example, students may be presented with a situation in which Naomi" – I think the solution should be given, in prose or a footnote: 3048:"The basic idea of the even more general" – I would drop "basic", and "first conceived" in the same sentence should just be "conceived". 2350:"A set is a collection of elements" – I think "unordered" and "distinct elements" needs to be said somewhere in the definition of "set". 1115:

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 (

4155: 3412:

The images are all free and strike a good balance of illustrating concepts and providing historical information. I'm not 100% sold that

122: 104: 33: 5765: 5206: 4395:

technical knowledge in algebra. Nominator only has 18 nominations and does not need a 2nd QPQ at this time (20 nominations required).

1666: 4541: 4424: 2761:

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

2121:

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.

377: 338: 5644:); keep only textbooks with a large audience and many citations, and specialized sources that are needed for some technical points. 5780: 5760: 304: 5301:

This is obviously only the barest introduction, but you'll notice that our article here does not even mention the concept of a

3070:

Most of them but I think not all. The formulation "Closely related developments" is intentionally vague to not imply too much.

1062: 110: 5729:– ideally, especially for a featured article, everything should be supported by the best, most relevant source we can find. – 4308: 665: 1966:) and "equations" (inequalities are also of interest, as are algorithms like "simplifying an expression" that go beyond the 1039: 2834: 5719:"The applications section has currently about 1000 words. Expanding it 20 times would be 20000 words for a single section. 5593:

section, shortening most of the part currently there and dramatically expanding the portion about the past century or two.

1662: 648:

The etymology relates algebra to transformation of equations. Indeed, until the 19th century, algebra was essentially the

4003: 652:; that is, the art of manipulating equations in view of solving them. This explain why some earlier work such as that of 573:

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

4813: 4805: 4316: 4123: 1106: 855: 313: 4098: 5346: 5306: 4171: 3964: 3646: 1825: 504: 5608:. If there's going to be such a section, recruit some mathematicians to brainstorm a list of applications of algebra. 539:

It might further be a good idea to expand the article in a few directions (while still staying within the limits of

5404:"Homomorphisms are tools in universal algebra to examine structural features by comparing two algebraic structures" 5357:

is covered in our article in the discussions of subalgebra while subrings and field extensions are also mentioned.

4829: 2495:

I implemented the second part. For the sake of simplicity, I think it's better to just stick with "equation" here.

1878: 5209:. But I'm not convinced that "major branches" is the most useful top-level organization system for this article. – 4872: 5575:

Make a top-level section about the main branches / fields of study in algebra. Substantially reduce the current

4022:

Research Issues in the Learning and Teaching of Algebra: The Research Agenda for Mathematics Education, Volume 4

2957:"in the form of a theory of equations" – I think this phrase needs explaining, at the very least in a footnote. 483: 4483: 4285: 3755: 3413: 2114:

Is it possible to say a bit more on the abstract algebra side about how numbers themselves are generalized? In

261: 71: 1228: 831: 3570:

I've made the corresponding adjustments. I appreciate all the time and effort you have put into this review!

1535:{\displaystyle \cos 3\theta {\text{cosec}}\theta +{\frac {\cos \theta }{\sin 3\theta }}\equiv 2\tan 2\theta } 851: 553: 5447: 5363: 5184: 4611: 4545: 4181: 955: 881: 805: 727: 2640: 4345: 4256: 4245: 2309: 2088: 1765: 1081: 471: 466: 461: 169: 5070:. Pratt 2022 is obviously not the only source to consider but it is one high-quality source to consider. 5406:

doesn't seem like a good way of introducing this topic), and as I said previously we have no mention of

5167:

We have sources that the first four subsection of the section "Major branches" are indeed major branches

5033: 5012: 4789: 4733: 4601: 4519: 3734: 3134: 1658: 319: 236: 92: 5237:

The division of algebra into elementary, linear, and abstract algebra is widely used. For example, see

754: 363: 5205:

If you want to decide what "major branches" are you could pull all of the algebra-related topics from

5675: 5641: 4809: 4801: 4781: 4721: 4439: 4400: 4312: 3032:. The basic idea here is that operations on vectors were conceived in terms of algebraic structures. 2115: 1967: 677: 657: 597: 586: 1641:

to both questions, which kind of creates a paradox. I added a short footnote to clarify this point.

977: 903: 5733: 5616: 5432: 5336: 5317: 5276: 5226: 5213: 5058: 4985: 4938: 4836: 4526: 4479: 4455: 2596: 2092: 2034: 1981: 1963: 1605: 981: 907: 902:

Provide a link to "coefficients" under the "Linear Algebra" section. 2nd paragraph, third sentence

669: 649: 221: 140: 1547:

three definitions the articles gives of "elementary algebra" (where another word may fit better).

750: 664:). During the 19th century, most of the older problems of algebra were solved by the proof of the 385:

on Knowledge. If you would like to participate, please visit the project page, where you can join

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to layperson readers or interested learners. Math encyclopediae and books for laypeople like the

2233: 1917: 1369: 1308: 1120: 686: 608: 540: 489: 369: 5242: 5011:

As explained above, Pratt 2022 is not reliable for anything. Terra 2018 is a textbook, and, per

353: 332: 2550: 5722: 5693:"the scope of graduate algebra courses is misguided since Knowledge is a general encyclopedia" 5683: 5512: 5483: 5456: 5407: 5374: 5258: 5246: 5238: 5157: 5127: 5075: 5041: 5002: 4969: 4914: 4891: 4861: 4785: 4753: 4717: 4653: 4638: 4575: 4469: 4199: 4146: 4055: 4026: 4020: 3970: 3832: 3592:. Thanks for all your work in this review, too. It's an extraordinary achievement to get this 3575: 3509: 3449: 3372: 3263: 3244: 3225: 3176: 3120: 3094: 3075: 3056: 3037: 3007: 2984: 2965: 2946: 2927: 2900: 2881: 2815: 2792: 2769: 2750: 2727: 2708: 2689: 2500: 2481: 2462: 2423: 2404: 2385: 2358: 2339: 2320: 2297: 2267: 2244: 2221: 2126: 2103: 2062: 2015: 1951: 1944:

Algebra ... seeks to solve equations ... abstract algebra is interested in such question as...

1894: 1865: 1836: 1788: 1776: 1719: 1678: 1646: 1611: 1589: 1556: 1444: 1316: 1155: 779: 701: 623: 563: 533: 529: 114: 4140: 3924: 2152: 1933: 5390:

from across the breadth of the subject, and we currently are barely scratching the surface.

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Kvasz, L. (2006). "The History of Algebra and the Development of the Form of Its Language".

3711: 3602: 3551: 3525: 3484: 3431: 3277: 3022: 2828: 2681: 2543:: a key idea is the inverse of an operation or the inverse of an order of steps (to isolate 2142: 2045: 2003: 1940: 1921: 1821: 1702: 1572: 1171: 1133: 1124: 1100: 1085: 485: 444: 188: 2181: 1334: 1193: 32:. A featured article should exemplify Knowledge's best work, and is therefore expected to 5721:– yes, you can see my point then: the current section is completely contrary to NPOV (cf. 5222: 5174: 4887: 4825: 4817: 4797: 4793: 4741: 4737: 4725: 4689: 4671: 4435: 4396: 4218: 3666: 3593: 2976:

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

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In note (a), "an algebraic operation is mapping" should be "is a mapping", but would "

5749: 5714:"earlier developments may seem simple from our point of view but they were important" 5649: 5535: 5493: 5192: 5138: 5107: 5016: 4942: 4895: 4883: 4821: 4729: 4618: 4558: 4478:

Agreed, it is very well-known that Arabic contributions to algebra were significant.

3963:

Maddocks, J. R. (2008). "Algebra". In Lerner, Brenda Wilmoth; Lerner, K. Lee (eds.).

2008:

If its subject is definable, then the first sentence should give a concise definition

1003: 929: 682: 673: 661: 604: 590: 3631:

The following is an archived discussion of the DYK nomination of the article below.

3021:

is a disambiguation link and also has unclear meaning (is this the invention of the

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series are suitable here as we only want an overview of an enormously broad topic.

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earliest" are hard to be sure of and often turn out to be incorrect. I note that

4415:

obvious. Plus the alphabet is used for the equations and not the Arabic script.

4390: 5678:

is about the content of articles, not about the sources to support that content.

4994: 2285:

Not all variables are maths formatted, like "x-y-pairs" without italics, and the

2259:

of lead, can "Algebra is relevant to ..." be merged with the previous paragraph?

5399: 4773: 4697: 3597: 3565: 3546: 3520: 3499: 3479: 3466: 3426: 2739: 2040: 2030: 1977: 1848: 1697: 1567: 1166: 1147: 1128: 1096: 382: 60: 5015:, it cannot be used as an example for the organization of a Knowledge article. 4924:

We can't go too much into depth and have to leave the details to child articles

4385: 3637:

Subsequent comments should be made on the appropriate discussion page (such as

556: 5170: 4705: 4701: 4666: 3995: 3707: 3663: 2084: 653: 582: 544: 359: 4518:. What we call "algebra" today, i.e. pushing alphabetic symbols around using 4434:

Umm... can you clarify what does that means? I cannot comprehend your words.

4115: 4081: 5425: 5187:

and sources can be found in the main articles and the linked article. Also,

4769: 4550: 4496:

For what it's worth, there are some other point of confusion here. The word

4090: 550: 230: 20: 4185: 3367:

Done. I slightly modified the example so that x represents Naomi's apples.

3067:"Closely related developments" – were these all 20th-century developments? 1088:. The edit link for this section can be used to add comments to the review. 5267:

You could alternately take the "branches of mathematics" section from the

1409:

axis graph); arbitrary numbers of variables typically use subscripts like

2256: 2083:" is often used for maths jargon). How about: "It is a generalization of 1987:

Or: the first sentence doesn't necessarily have to be a definition e.g. "

1604:

Is abstract algebra only about binary operations? Am I right in saying a

547: 5422:

Most people only ever get in contact with elementary and linear algebra.

5633: 5629: 5605: 5597: 5590: 5583: 5579:

section and put it in there (or it could frankly just be cut entirely).

5576: 5565: 5561: 5553: 5387: 4930: 4506: 3749: 3729: 3702: 2289:

is its own thing, which needs fixing in phrases like "either 2 or -2".

1937:

An important part of algebra, linear algebra, studies linear spaces,...

641: 574: 255: 5727:"important point is that everything is supported by a reliable source" 2331:

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

5532:

WT:WPM#Should Algebra be reverted to the version of 21 Decembre 2023?

4879:

For example, the section "Abstract algebra" already discusses groups.

4693: 4176: 3649:), unless there is consensus to re-open the discussion at this page. 3220:

I reformulated the sentence to not imply that this is the only goal.

2213:(but is changing the subject of a formula the same as "solving"?). 487: 3733:

methods were already studied in the ancient period even though an

1999:

mathematical objects but they are not directly studied by algebra.

1714:

I added a footnote to cover the different classification systems.

2039:

I see what you mean about my first suggestion being too broad. —

5708: 1791:

with its own idea of "balancing"; to express physical laws like

4592:

Knowledge:Featured article candidates/Algebra/archive1#D.Lazard

5736: 5687: 5657: 5619: 5604:

at the depth of discussion of the current content there about

5600:

section, which would need to be expanded probably 20x to meet

5543: 5516: 5501: 5487: 5460: 5435: 5378: 5339: 5320: 5279: 5262: 5216: 5200: 5161: 5146: 5131: 5115: 5079: 5061: 5045: 5024: 5006: 4988: 4973: 4950: 4918: 4903: 4865: 4839: 4678: 4657: 4642: 4626: 4579: 4564: 4529: 4487: 4473: 4458: 4443: 4428: 4404: 4203: 3818:, Lead Section, § 1. Elementary Algebra, § 2. Abstract Algebra 3669: 3609: 3579: 3558: 3532: 3513: 3491: 3453: 3438: 3376: 3267: 3248: 3229: 3124: 3098: 3079: 3060: 3041: 3011: 2988: 2969: 2950: 2931: 2904: 2885: 2819: 2796: 2773: 2754: 2731: 2712: 2693: 2504: 2485: 2466: 2427: 2408: 2389: 2362: 2343: 2324: 2301: 2271: 2248: 2225: 2130: 2107: 2066: 2052: 2019: 1955: 1898: 1869: 1840: 1780: 1723: 1709: 1682: 1673:

it rely on the people who use it to classify items correctly?

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on the table of content is much easier than word searching).

3754:

originates from an Arabic term for the surgical treatment of

260:

is derived from an Arabic term for the surgical treatment of

5701:"division into main branches creates an accessible overview" 4617:

tags, and sometimes a few worda on their main achievements.

4278:

Article is sourced, neutral, and free of copyright problems

19: 2037:

and the manipulation of statements within those structures.

1970:

properties of equations). Would it be fair to say instead:

5106:

lines, while remaing comprehensible for a large audience.

3769:

Template:Did you know nominations/Alfred de Laage de Meux

2996: 1696:

sort books by whether or not they are about "Algebra". —

4225:

the nomination until the hook appears on the Main Page.

3887:

harvnb error: no target: CITEREFMerzlyakovShirshov2020 (

3690: 3681: 200: 181: 162: 42: 5776:

Knowledge Did you know articles that are good articles

3898:, § Viète and the Formal Equation, § Analytic Geometry 2999:

justified (as a way to learn more about its history)?

2995:"the use of zero and negative numbers" – is a link to 2864:{\displaystyle \langle \mathbb {Z} ,+,\times \rangle } 2145:/overly specific link that could just be dropped e.g. 2079:

I don't like "regular numbers" at all (especially as "

5625:

I essentially agree, and would add a few more points

4331: 4271: 4231: 3737:

was not developed until the 16th and 17th centuries?

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I reformulated this passage to make it more concise.

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all the claims within this section also need sources

4512:

al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah

1584:

Ah, I see. I manage to reformulated those passages.

585:'s work is presented as algebra without saying that 381:, a collaborative effort to improve the coverage of 4338:Hook has been verified by provided inline citation 940:

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

656:were considered as belonging to algebra (presently 5414:. As another example, a one-sentence summary like 3882: 3836:harvnb error: no target: CITEREFWagnerKieran2018 ( 3356: 3327: 3301: 3209: 3165: 2863: 2668: 2629: 2581: 2535: 2446: 2205: 2170: 1812: 1756: 1626: 1534: 1428: 1393: 1358: 1294: 1217: 5440:I would oppose limiting the chemistry article to 4544:Thank you for your contribution, but I've undone 4002:. Metaphysics Research Lab, Stanford University. 1637:According to the reliable sources, the answer is 2454:an important enough inequation to be mentioned? 65:update the nomination page and article talk page 5801:Knowledge level-3 vital articles in Mathematics 3928:harvnb error: no target: CITEREFCresswell2010 ( 3025:and associated notation and basic operations?) 866:Semi-protected edit request on 21 February 2024 790:Semi-protected edit request on 17 February 2024 5663:general encyclopedia, not a math encyclopedia. 5442:most people's direct experience with chemistry 4449:nothing more than translate and preserve texts 4419:Diophantus and Theon point in this direction. 3807:harvnb error: no target: CITEREFMaddocks2008 ( 1692:are significant to mention as they imply that 712:Semi-protected edit request on 2 February 2024 120:If it no longer meets these criteria, you can 5548:Organization/scope changes I would consider: 1669:doesn't separate "algebra" as its own thing. 498:This page has archives. Sections older than 8: 4089:Merzlyakov, Yu. I.; Shirshov, A. I. (2020). 3861:harvnb error: no target: CITEREFTanton2005 ( 3828: 3651:No further edits should be made to this page 3471: 2858: 2838: 676:(characterisation of the equations that are 4500:comes from the name Al-Khwārizmī. The word 3940:harvnb error: no target: CITEREFOUP_Staff ( 3900:harvnb error: no target: CITEREFCorry2024 ( 3874:harvnb error: no target: CITEREFKvasz2006 ( 3820:harvnb error: no target: CITEREFPratt2022 ( 1017: 327: 134: 87: 5697:"current title is much more descriptive" 3920: 3340: 3314: 3279: 3178: 3144: 3136: 2842: 2841: 2836: 2662: 2642: 2603: 2598: 2561: 2552: 2516: 2439: 2380:I reformulated it to clarify this point. 2183: 2154: 1796: 1740: 1613: 1485: 1474: 1460: 1420: 1414: 1371: 1336: 1295:{\displaystyle (x-a)^{2}+(y-b)^{2}=r^{2}} 1286: 1273: 1248: 1230: 1195: 5705:"would overemphasize recent discoveries" 4019:Wagner, Sigrid; Kieran, Carolyn (2018). 3802: 1881:that could explain this. Or: is it only 5791:Knowledge vital articles in Mathematics 5384:"they could have chosen different ones" 5355:Creating New Structures Out of Old Ones 5051:The Stanford Encyclopedia of Philosophy 4410:Stop misappropriating facts and history 4000:The Stanford Encyclopedia of Philosophy 3792: 1048: 1020: 329: 5806:GA-Class vital articles in Mathematics 5726: 5718: 5713: 5704: 5700: 5696: 5692: 5530:I have opened a related discussion at 5421: 5391: 5383: 5359:Functions between Algebraic Structures 5311:classification of finite simple groups 5291: 5180: 5166: 4929:By the way, I had look on the section 4923: 4878: 4448: 4238:Article is new enough and long enough 3856: 3105: 2826: 2669:{\displaystyle -2,\times 5,{\sqrt {}}} 2025: 2007: 1943: 1936: 647: 508:when more than 5 sections are present. 5756:Knowledge featured article candidates 4514:) which was translated into Latin as 4421:2A02:85F:F80C:D100:9C0:4C78:DDF4:7AC1 4214:. Nominator has 18 past nominations. 3936: 3895: 3869: 3815: 1984:and the manipulation of such objects. 1331:; an equation of a line is typically 7: 5596:Eliminate or completely rewrite the 5351:Groups, Fields, Vector Spaces, Rings 5284:Here's what the introduction of the 5068:of particular interest to philosophy 4126:from the original on 19 January 2024 4006:from the original on 29 January 2024 3585: 3469:at the next stage. This is formally 1634:) and a 1-ary operation (the norm)? 375:This article is within the scope of 290: 288: 4194:Improved to Good Article status by 3166:{\displaystyle y={\frac {2}{5}}x+3} 2369:It's not clear from the prose that 668:(existence of complex solutions of 318:It is of interest to the following 5269:Princeton Companion to Mathematics 5207:Mathematics Subject Classification 1667:Mathematics Subject Classification 51:After one of the FAC coordinators 14: 5816:Top-priority mathematics articles 4142:Oxford Dictionary of Word Origins 4101:from the original on 7 April 2023 1690:Library of Congress QA 150-272.5 502:may be automatically archived by 395:Knowledge:WikiProject Mathematics 113:. If you can improve it further, 5786:Knowledge level-3 vital articles 4389: 4384: 4357: 4349: 4320: 4300: 4289: 4260: 4249: 3966:The Gale Encyclopedia of Science 3584: 3470: 992: 947: 918: 873: 842: 797: 765: 719: 443: 398:Template:WikiProject Mathematics 362: 352: 331: 298: 289: 220: 139: 91: 5796:GA-Class level-3 vital articles 5183:: Sure, but this is written in 4504:comes from Al-Khwārizmī's book 4158:from the original on 2024-01-27 4038:from the original on 2024-01-12 3982:from the original on 2024-01-12 3634:Please do not modify this page. 2292:Done. I hope I got everything. 640:I suggest to add at the end of 415:This article has been rated as 252:Did you know ... that the word 5658:20:44, 30 September 2024 (UTC) 5620:17:13, 30 September 2024 (UTC) 5544:15:43, 30 September 2024 (UTC) 5517:11:07, 30 September 2024 (UTC) 5502:17:04, 29 September 2024 (UTC) 5488:15:53, 29 September 2024 (UTC) 5436:19:11, 30 September 2024 (UTC) 5379:16:29, 30 September 2024 (UTC) 5340:14:41, 30 September 2024 (UTC) 5321:14:52, 30 September 2024 (UTC) 5280:14:23, 30 September 2024 (UTC) 5263:07:55, 30 September 2024 (UTC) 5217:18:19, 29 September 2024 (UTC) 5201:17:00, 29 September 2024 (UTC) 5162:15:43, 29 September 2024 (UTC) 5147:12:58, 29 September 2024 (UTC) 5132:12:05, 29 September 2024 (UTC) 5116:10:46, 29 September 2024 (UTC) 5080:07:40, 30 September 2024 (UTC) 5062:18:23, 29 September 2024 (UTC) 5046:15:49, 29 September 2024 (UTC) 5025:08:50, 29 September 2024 (UTC) 5007:08:24, 29 September 2024 (UTC) 4989:21:13, 28 September 2024 (UTC) 4974:08:17, 29 September 2024 (UTC) 4951:17:38, 28 September 2024 (UTC) 4919:16:25, 28 September 2024 (UTC) 4904:10:59, 28 September 2024 (UTC) 4866:07:49, 28 September 2024 (UTC) 4840:23:25, 27 September 2024 (UTC) 4679:22:00, 27 September 2024 (UTC) 4658:15:35, 27 September 2024 (UTC) 4643:15:33, 27 September 2024 (UTC) 4627:15:01, 27 September 2024 (UTC) 3883:Merzlyakov & Shirshov 2020 2630:{\displaystyle ^{2},\div 5,+2} 2149:is isolated in the inequality 1929:aim to familiarize students". 1270: 1257: 1245: 1232: 666:fundamental theorem of algebra 101:has been listed as one of the 1: 5811:GA-Class mathematics articles 4586:Incompleteness of the article 2960:I reformulated the sentence. 1008:16:33, 21 February 2024 (UTC) 986:16:26, 21 February 2024 (UTC) 934:16:33, 21 February 2024 (UTC) 912:16:18, 21 February 2024 (UTC) 860:09:50, 17 February 2024 (UTC) 836:09:20, 17 February 2024 (UTC) 552:, and the algebra entries of 389:and see a list of open tasks. 67:. Do not manually update the 5771:Old requests for peer review 4814:singular value decomposition 4806:eigenvalues and eigenvectors 4516:Liber Algebrae et Almucabola 4217:Post-promotion hook changes 3089:I simplified this sentence. 2312:" be better than "mapping"? 1542:, though I don't know where 1394:{\displaystyle A=bC+\theta } 784:20:36, 2 February 2024 (UTC) 759:19:54, 2 February 2024 (UTC) 706:17:04, 27 January 2024 (UTC) 691:15:34, 27 January 2024 (UTC) 628:13:32, 12 January 2024 (UTC) 613:12:11, 12 January 2024 (UTC) 568:14:03, 11 January 2024 (UTC) 5737:14:16, 1 October 2024 (UTC) 5688:08:03, 1 October 2024 (UTC) 5461:08:11, 1 October 2024 (UTC) 5347:insolubility of the quintic 5307:insolubility of the quintic 5229:, who has pointed out that 4368: 4221:on the talk page; consider 4095:Encyclopedia of Mathematics 4052:Encyclopedia of Mathematics 3872:, pp. 291–292, 297–298, 302 3647:Knowledge talk:Did you know 3639:this nomination's talk page 3596:to such a high standard. — 2807:might make a good example. 2511:On isolating variables and 2257:recommended four paragraphs 2024:I'd be happy enough to see 1995:concept that includes ..." 1879:concrete-pictorial-abstract 1608:has two binary operations ( 970:to reactivate your request. 958:has been answered. Set the 896:to reactivate your request. 884:has been answered. Set the 820:to reactivate your request. 808:has been answered. Set the 749:but a nit none the lesser. 742:to reactivate your request. 730:has been answered. Set the 250:The text of the entry was: 5832: 5554:§ Definition and etymology 5402:are or what they are for ( 5247:Diedrich & Lovett 2022 4830:discrete Fourier transform 4580:16:01, 8 August 2024 (UTC) 4565:15:54, 8 August 2024 (UTC) 4405:05:14, 25 March 2024 (UTC) 4204:12:39, 17 March 2024 (UTC) 3969:(4th ed.). Thompson Gale. 3735:abstract symbolic notation 3610:09:56, 17 March 2024 (UTC) 3580:09:42, 17 March 2024 (UTC) 3559:21:35, 16 March 2024 (UTC) 3533:16:50, 13 March 2024 (UTC) 3514:09:12, 13 March 2024 (UTC) 3454:09:40, 17 March 2024 (UTC) 3439:21:35, 16 March 2024 (UTC) 3377:09:00, 13 March 2024 (UTC) 3268:09:00, 13 March 2024 (UTC) 3249:09:00, 13 March 2024 (UTC) 3230:09:00, 13 March 2024 (UTC) 3125:09:00, 13 March 2024 (UTC) 3106:the connective "if...then" 3099:09:00, 13 March 2024 (UTC) 3080:18:35, 12 March 2024 (UTC) 3061:18:35, 12 March 2024 (UTC) 3042:18:35, 12 March 2024 (UTC) 3012:18:35, 12 March 2024 (UTC) 2989:18:35, 12 March 2024 (UTC) 2970:18:35, 12 March 2024 (UTC) 2951:18:35, 12 March 2024 (UTC) 2932:18:35, 12 March 2024 (UTC) 2905:17:53, 12 March 2024 (UTC) 2886:17:53, 12 March 2024 (UTC) 2820:17:53, 12 March 2024 (UTC) 2797:17:53, 12 March 2024 (UTC) 2774:17:53, 12 March 2024 (UTC) 2755:17:53, 12 March 2024 (UTC) 2732:13:47, 12 March 2024 (UTC) 2713:13:47, 12 March 2024 (UTC) 2694:13:47, 12 March 2024 (UTC) 2582:{\displaystyle 3x^{2}+2=c} 2505:13:47, 12 March 2024 (UTC) 2486:13:47, 12 March 2024 (UTC) 2467:13:25, 12 March 2024 (UTC) 2428:13:25, 12 March 2024 (UTC) 2409:13:25, 12 March 2024 (UTC) 2390:13:25, 12 March 2024 (UTC) 2363:13:25, 12 March 2024 (UTC) 2344:13:25, 12 March 2024 (UTC) 2325:12:49, 12 March 2024 (UTC) 2302:12:49, 12 March 2024 (UTC) 2272:12:30, 12 March 2024 (UTC) 2249:12:30, 12 March 2024 (UTC) 2226:12:27, 12 March 2024 (UTC) 2131:12:26, 12 March 2024 (UTC) 2108:10:02, 12 March 2024 (UTC) 2067:09:32, 17 March 2024 (UTC) 2053:21:35, 16 March 2024 (UTC) 2020:09:59, 12 March 2024 (UTC) 1980:that encompasses abstract 1956:09:32, 12 March 2024 (UTC) 1899:09:20, 12 March 2024 (UTC) 1870:18:38, 11 March 2024 (UTC) 1841:17:57, 11 March 2024 (UTC) 1781:17:00, 11 March 2024 (UTC) 1724:09:29, 17 March 2024 (UTC) 1710:21:35, 16 March 2024 (UTC) 1683:16:54, 11 March 2024 (UTC) 1651:16:45, 11 March 2024 (UTC) 1594:08:57, 17 March 2024 (UTC) 1580:21:35, 16 March 2024 (UTC) 1561:14:02, 11 March 2024 (UTC) 1449:13:35, 11 March 2024 (UTC) 1321:12:45, 11 March 2024 (UTC) 1160:07:58, 10 March 2024 (UTC) 672:) and the introduction of 642:§ Definition and etymology 30:featured article candidate 28:This article is a current 5766:Mathematics good articles 4488:20:42, 25 July 2024 (UTC) 4210:Number of QPQs required: 4139:Cresswell, Julia (2010). 3670:13:35, 2 April 2024 (UTC) 3492:23:33, 9 March 2024 (UTC) 3444:Done, I moved the image. 3210:{\displaystyle ax+by+c=0} 3108:– I think the spacing of 2637:done to it so we perform 2236:of algebraic structures? 1627:{\displaystyle +,\times } 1179:23:33, 9 March 2024 (UTC) 1141:19:41, 9 March 2024 (UTC) 1111:19:41, 9 March 2024 (UTC) 414: 347: 326: 271: 217: 137: 133: 105:Mathematics good articles 4756:, all of the objects of 4555:sold a copy of this book 4530:17:22, 9 June 2024 (UTC) 4474:07:51, 9 June 2024 (UTC) 4459:23:20, 8 June 2024 (UTC) 4444:13:27, 8 June 2024 (UTC) 4429:11:54, 8 June 2024 (UTC) 3829:Wagner & Kieran 2018 3414:File:Venn A subset B.svg 2913:From "Universal algebra" 2280:Up to "Abstract algebra" 2171:{\displaystyle b\leq -3} 636:Definition and etymology 421:project's priority scale 229:appeared on Knowledge's 5781:GA-Class vital articles 5761:Knowledge good articles 5448:Knowledge:Summary style 5364:Knowledge:Summary style 4542:Bjørn Leon Søren Riedel 4182:Oxford University Press 4120:Encyclopædia Britannica 4070:Philosophia Mathematica 3994:Pratt, Vaughan (2022). 3643:the article's talk page 3624:Did you know nomination 3405:I've been spotchecking 3395:Very Short Introduction 3302:{\displaystyle 2x+x=12} 2371:The Compendious Book... 378:WikiProject Mathematics 5568:into the same section. 5560:Eliminate the current 4082:10.1093/philmat/nkj017 4050:Tanton, James (2005). 3358: 3329: 3303: 3211: 3167: 3028:I disambiguated it to 2865: 2831:is a ring of the form 2670: 2631: 2583: 2537: 2448: 2373:is alternately titled 2207: 2206:{\displaystyle b=cx+d} 2172: 1824:(with its own idea of 1814: 1766:arithmetic progression 1758: 1628: 1536: 1430: 1395: 1360: 1359:{\displaystyle y=mx+c} 1296: 1219: 1218:{\displaystyle y=mx+c} 515:Changes to the article 505:Lowercase sigmabot III 227:fact from this article 24: 4790:representation theory 4734:cyclotomic polynomial 4520:mathematical notation 4202:). Self-nominated at 3885:, § Historical Survey 3385:Referencing and other 3359: 3330: 3304: 3212: 3168: 2866: 2671: 2632: 2584: 2538: 2536:{\displaystyle x-7=4} 2449: 2447:{\displaystyle \neq } 2208: 2173: 1815: 1759: 1629: 1537: 1431: 1429:{\displaystyle x_{i}} 1396: 1361: 1297: 1220: 658:Diophantine equations 589:belongs presently to 305:level-3 vital article 111:good article criteria 23: 5446:Bird's-eye view per 5032:As I understand it, 4810:matrix decomposition 4802:Gaussian elimination 4782:equivalence relation 4722:Diophantine equation 4536:Somma di arithmetica 4309:copyright violations 3746:: ... that the word 3700:... that elementary 3357:{\displaystyle 2x=8} 3339: 3313: 3278: 3177: 3135: 2835: 2641: 2597: 2551: 2515: 2438: 2182: 2153: 2116:algebraic structures 2093:algebraic operations 2035:algebraic structures 1982:mathematical objects 1968:equivalence relation 1813:{\displaystyle F=ma} 1795: 1757:{\displaystyle an+b} 1739: 1612: 1459: 1413: 1370: 1335: 1229: 1194: 826:s/happend/happened/ 678:solvable in radicals 670:polynomial equations 598:Gaussian elimination 587:Diophantine equation 401:mathematics articles 189:Good article nominee 170:Good article nominee 38:Please feel free to 5577:§ Universal algebra 5408:equivalence classes 5288:says about Algebra: 5286:Princeton Companion 4956:if you look at the 4939:weather forecasting 4114:Corry, Leo (2024). 3328:{\displaystyle x=4} 3112:should apply here. 1964:algebraic structure 1606:normed vector space 650:theory of equations 79:when the FAC closes 5634:§ Universal algbra 5630:§ Abstract algebra 5428:to those topics. – 5392:"are addressed by" 5243:Arcavi et al. 2017 4935:solving completely 4873:WP:tertiary source 4762:algebraic topology 4758:algebraic geometry 4317:close paraphrasing 3716:elementary algebra 3354: 3325: 3299: 3207: 3163: 2861: 2666: 2627: 2579: 2533: 2444: 2203: 2168: 1918:elementary algebra 1810: 1754: 1624: 1532: 1426: 1391: 1356: 1309:Algebraic geometry 1292: 1215: 1121:elementary algebra 852:Hyphenation Expert 370:Mathematics portal 314:content assessment 145:Article milestones 25: 5552:Possibly retitle 5451:, not exclusion. 5353:. The part about 5223:original research 4892:quantum mechanics 4786:equivalence class 4754:rational function 4718:geometric algebra 4379: 4378: 4367: 4366: 4330: 4329: 4286:Adequate sourcing 4270: 4269: 4207: 4151:978-0-19-954793-7 4060:978-0-8160-5124-3 4054:. Facts On File. 4031:978-1-135-43421-2 3975:978-1-4144-2877-2 3761: 3740: 3721: 3152: 2664: 2095:other than ..."? 2029:is the branch of 1976:is the branch of 1789:chemical equation 1663:Dewey Decimal 512 1512: 1477: 1076: 1075: 974: 973: 900: 899: 824: 823: 746: 745: 512: 511: 477: 476: 435: 434: 431: 430: 427: 426: 283: 282: 279: 278: 129: 86: 85: 82: 34:meet the criteria 5823: 5584:§ Linear algebra 5562:§ Major branches 5450: 5443: 5417: 5405: 5360: 5356: 5352: 5303:Euclidean domain 5233: 5191:may be applied. 5185:wp:summary style 5069: 4931:§ Linear algebra 4714:Clifford algebra 4710:exterior algebra 4676: 4674: 4669: 4616: 4610: 4606: 4600: 4393: 4388: 4369: 4361: 4360: 4353: 4352: 4332: 4324: 4323: 4304: 4303: 4293: 4292: 4272: 4264: 4263: 4253: 4252: 4232: 4193: 4189: 4184:. Archived from 4166: 4164: 4163: 4135: 4133: 4131: 4110: 4108: 4106: 4085: 4064: 4046: 4044: 4043: 4015: 4013: 4011: 3990: 3988: 3987: 3949: 3945: 3933: 3915: 3909: 3905: 3892: 3879: 3866: 3851: 3845: 3841: 3825: 3812: 3797: 3759: 3738: 3719: 3712:abstract algebra 3658:The result was: 3636: 3605: 3588: 3587: 3569: 3554: 3543: 3528: 3503: 3487: 3474: 3473: 3434: 3363: 3361: 3360: 3355: 3334: 3332: 3331: 3326: 3308: 3306: 3305: 3300: 3216: 3214: 3213: 3208: 3172: 3170: 3169: 3164: 3153: 3145: 3023:Euclidean vector 2979:No. I fixed it. 2870: 2868: 2867: 2862: 2845: 2829:ring of integers 2682:Equation solving 2675: 2673: 2672: 2667: 2665: 2663: 2636: 2634: 2633: 2628: 2608: 2607: 2588: 2586: 2585: 2580: 2566: 2565: 2542: 2540: 2539: 2534: 2453: 2451: 2450: 2445: 2255:To stick to the 2212: 2210: 2209: 2204: 2178:or the equation 2177: 2175: 2174: 2169: 2140: 2087:that introduces 2048: 2004:MOS:LEADSENTENCE 1922:abstract algebra 1822:computer science 1819: 1817: 1816: 1811: 1763: 1761: 1760: 1755: 1705: 1659:an exam syllabus 1633: 1631: 1630: 1625: 1575: 1541: 1539: 1538: 1533: 1513: 1511: 1497: 1486: 1478: 1475: 1435: 1433: 1432: 1427: 1425: 1424: 1400: 1398: 1397: 1392: 1365: 1363: 1362: 1357: 1301: 1299: 1298: 1293: 1291: 1290: 1278: 1277: 1253: 1252: 1224: 1222: 1221: 1216: 1174: 1136: 1125:abstract algebra 1086:Talk:Algebra/GA1 1030:Copyvio detector 1018: 1000: 996: 995: 965: 961: 951: 950: 944: 926: 922: 921: 891: 887: 877: 876: 870: 850: 846: 845: 815: 811: 801: 800: 794: 773: 769: 768: 737: 733: 723: 722: 716: 507: 491: 458: 457: 447: 439: 403: 402: 399: 396: 393: 372: 367: 366: 356: 349: 348: 343: 335: 328: 311: 302: 301: 294: 293: 292: 285: 272:Current status: 224: 203: 184: 165: 163:November 6, 2007 144: 143: 135: 118: 95: 88: 76: 70: 59:the nomination, 50: 48: 16: 5831: 5830: 5826: 5825: 5824: 5822: 5821: 5820: 5746: 5745: 5628:Merge sections 5445: 5441: 5415: 5403: 5358: 5354: 5350: 5230: 5225:. I agree with 5175:Moderne Algebra 5067: 4888:crystallography 4826:spectral theory 4818:polynomial ring 4798:linear subspace 4794:category theory 4742:lattice (group) 4738:lattice (order) 4726:field extension 4690:multilinear map 4672: 4667: 4665: 4614: 4608: 4604: 4598: 4588: 4538: 4412: 4407: 4358: 4350: 4321: 4301: 4290: 4261: 4250: 4169: 4161: 4159: 4152: 4138: 4129: 4127: 4113: 4104: 4102: 4088: 4067: 4061: 4049: 4041: 4039: 4032: 4018: 4009: 4007: 3993: 3985: 3983: 3976: 3962: 3954: 3953: 3952: 3948: 3939: 3927: 3916: 3912: 3908: 3899: 3886: 3873: 3860: 3852: 3848: 3844: 3835: 3819: 3806: 3798: 3794: 3697: 3695: 3691:Article history 3632: 3626: 3603: 3563: 3552: 3537: 3526: 3497: 3485: 3462: 3432: 3387: 3337: 3336: 3311: 3310: 3276: 3275: 3175: 3174: 3133: 3132: 2915: 2833: 2832: 2782:natural numbers 2639: 2638: 2600: 2595: 2594: 2557: 2549: 2548: 2513: 2512: 2436: 2435: 2282: 2180: 2179: 2151: 2150: 2138: 2046: 1914: 1909: 1793: 1792: 1737: 1736: 1703: 1610: 1609: 1573: 1498: 1487: 1457: 1456: 1416: 1411: 1410: 1368: 1367: 1333: 1332: 1282: 1269: 1244: 1227: 1226: 1192: 1191: 1186: 1172: 1134: 1080:This review is 1072: 1044: 1016: 993: 991: 963: 959: 948: 942: 919: 917: 889: 885: 874: 868: 843: 841: 813: 809: 798: 792: 766: 764: 735: 731: 720: 714: 638: 517: 503: 492: 486: 452: 400: 397: 394: 391: 390: 368: 361: 341: 312:on Knowledge's 309: 299: 267: 266: 248: 199: 180: 161: 138: 74: 72:Article history 68: 55:the article or 39: 12: 11: 5: 5829: 5827: 5819: 5818: 5813: 5808: 5803: 5798: 5793: 5788: 5783: 5778: 5773: 5768: 5763: 5758: 5748: 5747: 5744: 5743: 5742: 5741: 5740: 5739: 5672: 5664: 5647: 5646: 5645: 5638: 5610: 5609: 5598:§ Applications 5594: 5587: 5580: 5573: 5569: 5558: 5528: 5527: 5526: 5525: 5524: 5523: 5522: 5521: 5520: 5519: 5475: 5474: 5473: 5472: 5471: 5470: 5469: 5468: 5467: 5466: 5465: 5464: 5463: 5419: 5367: 5329: 5325: 5324: 5323: 5299: 5295: 5289: 5235: 5227:Mathwriter2718 5203: 5178: 5103: 5102: 5101: 5100: 5099: 5098: 5097: 5096: 5095: 5094: 5093: 5092: 5091: 5090: 5089: 5088: 5087: 5086: 5085: 5084: 5083: 5082: 5034:WP:NOTTEXTBOOK 5030: 5013:WP:NOTTEXTBOOK 4978: 4977: 4976: 4961: 4927: 4876: 4853: 4846: 4778:symmetry group 4746:discrete group 4686:complex number 4681: 4645: 4612:expand section 4590:As I noted in 4587: 4584: 4583: 4582: 4537: 4534: 4533: 4532: 4494: 4493: 4492: 4491: 4490: 4480:Mathwriter2718 4446: 4411: 4408: 4377: 4376: 4365: 4364: 4363: 4362: 4354: 4340: 4339: 4328: 4327: 4326: 4325: 4305: 4294: 4280: 4279: 4268: 4267: 4266: 4265: 4254: 4240: 4239: 4230: 4229: 4219:will be logged 4191: 4190: 4188:on 2013-11-20. 4167: 4150: 4145:. OUP Oxford. 4136: 4111: 4086: 4065: 4059: 4047: 4030: 4016: 3991: 3974: 3951: 3950: 3947: 3946: 3934: 3921:Cresswell 2010 3917: 3910: 3907: 3906: 3893: 3880: 3867: 3853: 3846: 3843: 3842: 3826: 3813: 3799: 3791: 3790: 3786: 3780: 3779: 3778: 3777: 3771: 3762: 3741: 3694: 3693: 3688: 3678: 3676: 3672: 3656: 3655: 3627: 3625: 3622: 3621: 3620: 3619: 3618: 3617: 3616: 3615: 3614: 3613: 3612: 3461: 3458: 3457: 3456: 3386: 3383: 3382: 3381: 3380: 3379: 3353: 3350: 3347: 3344: 3335:and Naomi has 3324: 3321: 3318: 3298: 3295: 3292: 3289: 3286: 3283: 3272: 3271: 3270: 3253: 3252: 3251: 3234: 3233: 3232: 3206: 3203: 3200: 3197: 3194: 3191: 3188: 3185: 3182: 3162: 3159: 3156: 3151: 3148: 3143: 3140: 3129: 3128: 3127: 3103: 3102: 3101: 3084: 3083: 3082: 3065: 3064: 3063: 3046: 3045: 3044: 3019:vector algebra 3016: 3015: 3014: 2993: 2992: 2991: 2974: 2973: 2972: 2955: 2954: 2953: 2936: 2935: 2934: 2914: 2911: 2910: 2909: 2908: 2907: 2890: 2889: 2888: 2860: 2857: 2854: 2851: 2848: 2844: 2840: 2824: 2823: 2822: 2805:symmetry group 2801: 2800: 2799: 2778: 2777: 2776: 2759: 2758: 2757: 2736: 2735: 2734: 2717: 2716: 2715: 2698: 2697: 2696: 2661: 2658: 2655: 2652: 2649: 2646: 2626: 2623: 2620: 2617: 2614: 2611: 2606: 2602: 2578: 2575: 2572: 2569: 2564: 2560: 2556: 2532: 2529: 2526: 2523: 2520: 2509: 2508: 2507: 2490: 2489: 2488: 2471: 2470: 2469: 2443: 2432: 2431: 2430: 2413: 2412: 2411: 2394: 2393: 2392: 2367: 2366: 2365: 2348: 2347: 2346: 2329: 2328: 2327: 2306: 2305: 2304: 2281: 2278: 2277: 2276: 2275: 2274: 2253: 2252: 2251: 2230: 2229: 2228: 2202: 2199: 2196: 2193: 2190: 2187: 2167: 2164: 2161: 2158: 2135: 2134: 2133: 2112: 2111: 2110: 2077: 2076: 2075: 2074: 2073: 2072: 2071: 2070: 2069: 2000: 1985: 1960: 1959: 1958: 1913: 1910: 1908: 1907:Prose comments 1905: 1904: 1903: 1902: 1901: 1874: 1873: 1872: 1845: 1844: 1843: 1809: 1806: 1803: 1800: 1785: 1784: 1783: 1753: 1750: 1747: 1744: 1732: 1731: 1730: 1729: 1728: 1727: 1726: 1694:most libraries 1655: 1654: 1653: 1623: 1620: 1617: 1602: 1601: 1600: 1599: 1598: 1597: 1596: 1531: 1528: 1525: 1522: 1519: 1516: 1510: 1507: 1504: 1501: 1496: 1493: 1490: 1484: 1481: 1473: 1470: 1467: 1464: 1453: 1452: 1451: 1423: 1419: 1390: 1387: 1384: 1381: 1378: 1375: 1355: 1352: 1349: 1346: 1343: 1340: 1325: 1324: 1323: 1289: 1285: 1281: 1276: 1272: 1268: 1265: 1262: 1259: 1256: 1251: 1247: 1243: 1240: 1237: 1234: 1214: 1211: 1208: 1205: 1202: 1199: 1185: 1184:Scope comments 1182: 1163: 1162: 1091: 1090: 1074: 1073: 1071: 1070: 1065: 1060: 1054: 1051: 1050: 1046: 1045: 1043: 1042: 1040:External links 1037: 1032: 1026: 1023: 1022: 1015: 1012: 1011: 1010: 1001: 972: 971: 952: 941: 938: 937: 936: 927: 898: 897: 878: 867: 864: 863: 862: 828:78.119.119.111 822: 821: 802: 791: 788: 787: 786: 744: 743: 724: 713: 710: 709: 708: 637: 634: 633: 632: 631: 630: 601: 594: 578: 516: 513: 510: 509: 497: 494: 493: 488: 484: 482: 479: 478: 475: 474: 469: 464: 454: 453: 448: 442: 433: 432: 429: 428: 425: 424: 413: 407: 406: 404: 387:the discussion 374: 373: 357: 345: 344: 336: 324: 323: 317: 295: 281: 280: 277: 276: 269: 268: 249: 219: 218: 215: 214: 211: 204: 201:August 6, 2024 196: 195: 192: 185: 182:March 17, 2024 177: 176: 173: 166: 158: 157: 154: 151: 147: 146: 131: 130: 96: 84: 83: 49: 43:leave comments 37: 26: 13: 10: 9: 6: 4: 3: 2: 5828: 5817: 5814: 5812: 5809: 5807: 5804: 5802: 5799: 5797: 5794: 5792: 5789: 5787: 5784: 5782: 5779: 5777: 5774: 5772: 5769: 5767: 5764: 5762: 5759: 5757: 5754: 5753: 5751: 5738: 5735: 5732: 5728: 5724: 5720: 5715: 5710: 5706: 5702: 5698: 5694: 5691: 5690: 5689: 5685: 5681: 5677: 5676:WP:NOTCATALOG 5673: 5670: 5665: 5661: 5660: 5659: 5655: 5651: 5648: 5643: 5642:WP:NOTCATALOG 5639: 5635: 5631: 5627: 5626: 5624: 5623: 5622: 5621: 5618: 5615: 5607: 5603: 5599: 5595: 5592: 5588: 5585: 5581: 5578: 5574: 5570: 5567: 5563: 5559: 5555: 5551: 5550: 5549: 5546: 5545: 5541: 5537: 5533: 5518: 5514: 5510: 5505: 5504: 5503: 5499: 5495: 5491: 5490: 5489: 5485: 5481: 5476: 5462: 5458: 5454: 5449: 5439: 5438: 5437: 5434: 5431: 5427: 5423: 5420: 5413: 5412:quotient maps 5409: 5401: 5397: 5396:homomorphisms 5393: 5389: 5385: 5382: 5381: 5380: 5376: 5372: 5368: 5365: 5348: 5343: 5342: 5341: 5338: 5335: 5330: 5326: 5322: 5319: 5316: 5312: 5308: 5304: 5300: 5298: 5294: 5290: 5287: 5283: 5282: 5281: 5278: 5275: 5270: 5266: 5265: 5264: 5260: 5256: 5252: 5248: 5244: 5240: 5236: 5228: 5224: 5220: 5219: 5218: 5215: 5212: 5208: 5204: 5202: 5198: 5194: 5190: 5186: 5182: 5179: 5176: 5172: 5168: 5165: 5164: 5163: 5159: 5155: 5150: 5149: 5148: 5144: 5140: 5135: 5134: 5133: 5129: 5125: 5120: 5119: 5118: 5117: 5113: 5109: 5081: 5077: 5073: 5065: 5064: 5063: 5060: 5057: 5052: 5049: 5048: 5047: 5043: 5039: 5035: 5031: 5028: 5027: 5026: 5022: 5018: 5014: 5010: 5009: 5008: 5004: 5000: 4996: 4992: 4991: 4990: 4987: 4984: 4979: 4975: 4971: 4967: 4962: 4959: 4954: 4953: 4952: 4948: 4944: 4940: 4936: 4932: 4928: 4925: 4922: 4921: 4920: 4916: 4912: 4907: 4906: 4905: 4901: 4897: 4893: 4889: 4885: 4880: 4877: 4874: 4869: 4868: 4867: 4863: 4859: 4854: 4851: 4847: 4843: 4842: 4841: 4838: 4835: 4831: 4827: 4823: 4822:Hilbert space 4819: 4815: 4811: 4807: 4803: 4799: 4795: 4791: 4787: 4783: 4779: 4775: 4771: 4767: 4766:Gröbner basis 4763: 4759: 4755: 4751: 4747: 4743: 4739: 4735: 4731: 4730:root of unity 4727: 4723: 4719: 4715: 4711: 4707: 4703: 4699: 4695: 4691: 4687: 4682: 4680: 4677: 4675: 4670: 4661: 4660: 4659: 4655: 4651: 4646: 4644: 4640: 4636: 4631: 4630: 4629: 4628: 4624: 4620: 4613: 4603: 4595: 4593: 4585: 4581: 4577: 4573: 4569: 4568: 4567: 4566: 4563: 4560: 4556: 4552: 4547: 4543: 4535: 4531: 4528: 4525: 4521: 4517: 4513: 4509: 4508: 4503: 4499: 4495: 4489: 4485: 4481: 4477: 4476: 4475: 4471: 4467: 4462: 4461: 4460: 4457: 4454: 4450: 4447: 4445: 4441: 4437: 4433: 4432: 4431: 4430: 4426: 4422: 4416: 4409: 4406: 4402: 4398: 4392: 4387: 4382: 4374: 4371: 4370: 4356:Interesting: 4355: 4347: 4344: 4343: 4342: 4341: 4337: 4334: 4333: 4318: 4314: 4310: 4306: 4298: 4295: 4287: 4284: 4283: 4282: 4281: 4277: 4274: 4273: 4258: 4255: 4247: 4244: 4243: 4242: 4241: 4237: 4234: 4233: 4228: 4227: 4226: 4224: 4220: 4215: 4213: 4208: 4205: 4201: 4197: 4187: 4183: 4179: 4178: 4173: 4168: 4157: 4153: 4148: 4144: 4143: 4137: 4125: 4121: 4117: 4112: 4100: 4096: 4092: 4087: 4083: 4079: 4075: 4071: 4066: 4062: 4057: 4053: 4048: 4037: 4033: 4028: 4025:. Routledge. 4024: 4023: 4017: 4005: 4001: 3997: 3992: 3981: 3977: 3972: 3968: 3967: 3961: 3960: 3959: 3958: 3943: 3938: 3935: 3931: 3926: 3922: 3919: 3918: 3914: 3911: 3903: 3897: 3894: 3890: 3884: 3881: 3877: 3871: 3868: 3864: 3858: 3855: 3854: 3850: 3847: 3839: 3834: 3830: 3827: 3823: 3817: 3814: 3810: 3805:, pp. 129–131 3804: 3803:Maddocks 2008 3801: 3800: 3796: 3793: 3789: 3785: 3784: 3775: 3772: 3770: 3766: 3763: 3757: 3753: 3752: 3751: 3745: 3742: 3736: 3732: 3731: 3726: 3723: 3722: 3717: 3713: 3709: 3705: 3704: 3699: 3698: 3692: 3689: 3687: 3683: 3680: 3679: 3675: 3673: 3671: 3668: 3665: 3661: 3654: 3652: 3648: 3644: 3640: 3635: 3629: 3628: 3623: 3611: 3607: 3606: 3599: 3595: 3594:vital article 3591: 3583: 3582: 3581: 3577: 3573: 3567: 3562: 3561: 3560: 3556: 3555: 3548: 3541: 3536: 3535: 3534: 3530: 3529: 3522: 3517: 3516: 3515: 3511: 3507: 3501: 3496: 3495: 3494: 3493: 3489: 3488: 3481: 3477: 3468: 3467:verifiability 3459: 3455: 3451: 3447: 3443: 3442: 3441: 3440: 3436: 3435: 3428: 3424: 3420: 3415: 3410: 3408: 3403: 3400: 3396: 3392: 3391:accessibility 3384: 3378: 3374: 3370: 3366: 3365: 3351: 3348: 3345: 3342: 3322: 3319: 3316: 3296: 3293: 3290: 3287: 3284: 3281: 3273: 3269: 3265: 3261: 3257: 3256: 3254: 3250: 3246: 3242: 3238: 3237: 3235: 3231: 3227: 3223: 3219: 3218: 3204: 3201: 3198: 3195: 3192: 3189: 3186: 3183: 3180: 3160: 3157: 3154: 3149: 3146: 3141: 3138: 3130: 3126: 3122: 3118: 3114: 3113: 3111: 3107: 3104: 3100: 3096: 3092: 3088: 3087: 3085: 3081: 3077: 3073: 3069: 3068: 3066: 3062: 3058: 3054: 3050: 3049: 3047: 3043: 3039: 3035: 3031: 3027: 3026: 3024: 3020: 3017: 3013: 3009: 3005: 3001: 3000: 2998: 2994: 2990: 2986: 2982: 2978: 2977: 2975: 2971: 2967: 2963: 2959: 2958: 2956: 2952: 2948: 2944: 2940: 2939: 2937: 2933: 2929: 2925: 2920: 2919: 2917: 2916: 2912: 2906: 2902: 2898: 2894: 2893: 2891: 2887: 2883: 2879: 2875: 2874: 2872: 2855: 2852: 2849: 2846: 2830: 2825: 2821: 2817: 2813: 2809: 2808: 2806: 2802: 2798: 2794: 2790: 2786: 2785: 2783: 2779: 2775: 2771: 2767: 2763: 2762: 2760: 2756: 2752: 2748: 2744: 2743: 2741: 2737: 2733: 2729: 2725: 2721: 2720: 2718: 2714: 2710: 2706: 2702: 2701: 2699: 2695: 2691: 2687: 2683: 2678: 2677: 2659: 2656: 2653: 2650: 2647: 2644: 2624: 2621: 2618: 2615: 2612: 2609: 2604: 2601: 2592: 2576: 2573: 2570: 2567: 2562: 2558: 2554: 2546: 2530: 2527: 2524: 2521: 2518: 2510: 2506: 2502: 2498: 2494: 2493: 2491: 2487: 2483: 2479: 2475: 2474: 2472: 2468: 2464: 2460: 2456: 2455: 2441: 2433: 2429: 2425: 2421: 2417: 2416: 2414: 2410: 2406: 2402: 2398: 2397: 2395: 2391: 2387: 2383: 2379: 2378: 2376: 2372: 2368: 2364: 2360: 2356: 2352: 2351: 2349: 2345: 2341: 2337: 2333: 2332: 2330: 2326: 2322: 2318: 2314: 2313: 2311: 2307: 2303: 2299: 2295: 2291: 2290: 2288: 2284: 2283: 2279: 2273: 2269: 2265: 2261: 2260: 2258: 2254: 2250: 2246: 2242: 2238: 2237: 2235: 2231: 2227: 2223: 2219: 2215: 2214: 2200: 2197: 2194: 2191: 2188: 2185: 2165: 2162: 2159: 2156: 2148: 2144: 2136: 2132: 2128: 2124: 2120: 2119: 2117: 2113: 2109: 2105: 2101: 2097: 2096: 2094: 2090: 2086: 2082: 2078: 2068: 2064: 2060: 2056: 2055: 2054: 2050: 2049: 2042: 2038: 2036: 2033:that studies 2032: 2028: 2023: 2022: 2021: 2017: 2013: 2009: 2005: 2002:According to 2001: 1997: 1996: 1994: 1990: 1986: 1983: 1979: 1975: 1972: 1971: 1969: 1965: 1961: 1957: 1953: 1949: 1945: 1941: 1938: 1934: 1931: 1930: 1927: 1926: 1925: 1923: 1919: 1911: 1906: 1900: 1896: 1892: 1887: 1886: 1884: 1880: 1875: 1871: 1867: 1863: 1859: 1858: 1855: 1850: 1846: 1842: 1838: 1834: 1830: 1829: 1827: 1823: 1807: 1804: 1801: 1798: 1790: 1786: 1782: 1778: 1774: 1770: 1769: 1767: 1764:generates an 1751: 1748: 1745: 1742: 1733: 1725: 1721: 1717: 1713: 1712: 1711: 1707: 1706: 1699: 1695: 1691: 1686: 1685: 1684: 1680: 1676: 1671: 1670: 1668: 1664: 1660: 1656: 1652: 1648: 1644: 1640: 1636: 1635: 1621: 1618: 1615: 1607: 1603: 1595: 1591: 1587: 1583: 1582: 1581: 1577: 1576: 1569: 1564: 1563: 1562: 1558: 1554: 1549: 1548: 1545: 1529: 1526: 1523: 1520: 1517: 1514: 1508: 1505: 1502: 1499: 1494: 1491: 1488: 1482: 1479: 1471: 1468: 1465: 1462: 1454: 1450: 1446: 1442: 1438: 1437: 1421: 1417: 1408: 1404: 1388: 1385: 1382: 1379: 1376: 1373: 1353: 1350: 1347: 1344: 1341: 1338: 1330: 1326: 1322: 1318: 1314: 1310: 1305: 1304: 1287: 1283: 1279: 1274: 1266: 1263: 1260: 1254: 1249: 1241: 1238: 1235: 1212: 1209: 1206: 1203: 1200: 1197: 1188: 1187: 1183: 1181: 1180: 1176: 1175: 1168: 1161: 1157: 1153: 1149: 1145: 1144: 1143: 1142: 1138: 1137: 1130: 1126: 1122: 1118: 1113: 1112: 1108: 1105: 1102: 1098: 1095: 1089: 1087: 1083: 1078: 1077: 1069: 1066: 1064: 1061: 1059: 1056: 1055: 1053: 1052: 1047: 1041: 1038: 1036: 1033: 1031: 1028: 1027: 1025: 1024: 1019: 1013: 1009: 1005: 999: 990: 989: 988: 987: 983: 979: 969: 966:parameter to 957: 953: 946: 945: 939: 935: 931: 925: 916: 915: 914: 913: 909: 905: 895: 892:parameter to 883: 879: 872: 871: 865: 861: 857: 853: 849: 840: 839: 838: 837: 833: 829: 819: 816:parameter to 807: 803: 796: 795: 789: 785: 781: 777: 772: 763: 762: 761: 760: 756: 752: 741: 738:parameter to 729: 725: 718: 717: 711: 707: 703: 699: 695: 694: 693: 692: 688: 684: 681: 679: 675: 674:Galois theory 671: 667: 663: 662:number theory 659: 655: 651: 645: 643: 635: 629: 625: 621: 616: 615: 614: 610: 606: 602: 599: 595: 592: 591:number theory 588: 584: 579: 576: 572: 571: 570: 569: 565: 561: 557: 554: 551: 548: 545: 542: 537: 535: 531: 525: 523: 514: 506: 501: 496: 495: 481: 480: 473: 470: 468: 465: 463: 460: 459: 456: 455: 451: 446: 441: 440: 437: 422: 418: 412: 409: 408: 405: 388: 384: 380: 379: 371: 365: 360: 358: 355: 351: 350: 346: 340: 337: 334: 330: 325: 321: 315: 307: 306: 296: 287: 286: 275: 270: 265: 263: 258: 257: 253: 246: 245: 244:April 8, 2024 240: 238: 237:Did you know? 232: 228: 223: 216: 212: 210: 209: 205: 202: 198: 197: 193: 191: 190: 186: 183: 179: 178: 174: 172: 171: 167: 164: 160: 159: 155: 152: 149: 148: 142: 136: 132: 127: 125: 124: 116: 112: 108: 107: 106: 100: 97: 94: 90: 89: 80: 73: 66: 62: 58: 54: 46: 45: 44: 35: 31: 27: 22: 18: 17: 5611: 5589:Refocus the 5547: 5529: 5400:isomorphisms 5292: 5285: 5268: 5239:O’Regan 2020 5104: 5050: 4934: 4884:Rubik's cube 4750:finite field 4664: 4602:main article 4596: 4589: 4553:, when they 4539: 4515: 4511: 4505: 4501: 4497: 4417: 4413: 4380: 4372: 4335: 4275: 4235: 4216: 4211: 4209: 4192: 4186:the original 4175: 4160:. 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