# Talk:Field (mathematics)

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Field:  Algebra

## Too many Operations?

I am pretty sure the definition of a field requires that the algebraic structure is endowed with exactly 2 operations. These operations are often called multiplication and addition. I think that the intro should emphasize that the operations might be very different than the addition and multiplication from the real numbers (for example, vector addition is very different). There is no need for subtraction and division as these 'operations' come from the presence of inverse elements with respect to the two operations. Are inverse elements same as corrective measures. BioPseudo (talk) 09:20, 28 January 2017 (UTC)

You're quite right and I edited the article accordingly. Vincent (talk) 17:29, 3 April 2017 (UTC)

Previous post moved outside the post inside which it was included. D.Lazard (talk) 22:09, 3 April 2017 (UTC)

The existence of subtraction and division results from the axioms, but providing a method for computing the multiplicative inverse is strictly equivalent to define it as a unary operation. Equivalently if one wants axiom with only ${\displaystyle \forall }$ and without ${\displaystyle \exists ,}$ one has to consider multiplicative inverses and additive inverses as unary operations. Thus, the number of operations of a field may be 2 or 4 depending on the definition which is chosen. D.Lazard (talk) 22:09, 3 April 2017 (UTC)

## A field is not a set

The introduction says 'it is a set on which are defined... ' This would have the reader believe that a field is a set. It is not a set. It is an algebraic structure.

Ditto: you're right and I edited. Vincent (talk) 17:29, 3 April 2017 (UTC)
You are wrong and I have reverted your edit. A field, as every math. structure, is a set equipped with operations. I know that some pedantic people say that a field is a triplet formed by a set and two operations. Beside being pedantic, this formulation makes incorrect a formula such as ${\displaystyle x\in F,}$ and does not correspond to the practice of people who work with fields. For example, for almost everybody, ${\displaystyle {\mathbb {Q}}}$ denote both the set and the field of rational numbers. To my knowledge, the only case where it may be useful to include the operations in the notation of a structure is when there are several structures of the same nature on the same set. This never happens for fields. Even when the operation appears in the notation, an algebraic structure has elements, is thus a set, and must be thought as a set on which the operations operate.
Moreover the manual of style say that technicalities must be avoided, as far as possible, in the lead, and if a formal presentation is needed, it is better to delay it to the body of the article. D.Lazard (talk) 21:52, 3 April 2017 (UTC)
Hello D.Lazard. I took account of your criticism and kept the initial line as simple as possible without making a false statement.
The fact is that a field is a set with two operations with each operation having an identity element and each element having an inverse with respect to each operation, except for the additive identity which has no multiplicative inverse, and the multiplicative identity which is its own inverse and does not have a distinct inverse. That is the very definition of a field.
I also wrote 1/${\displaystyle a}$ rather than ${\displaystyle a^{-1}}$ for the multiplicative inverse.
I hope we can agree on this as a consensus. If you do not agree, would you please consider making a constructive edit rather than simply reverting? Cheers! Vincent (talk) 15:22, 4 April 2017 (UTC)
I repeat my request for consensus and I invite you to behave in a more collegial manner. Rgds. Vincent (talk) 16:48, 4 April 2017 (UTC)

The lead seems overly technical. I move to simplify it significantly, as well as shuffle most of its present contents below in the main text. Inorout (talk) 12:27, 16 March 2015 (UTC)

An editor made several edits of the lead of the article, which I have reverted, because some did not reflected the common practice in mathematics, and the others do not follow the guidelines of MOS:MATH, which says "The lead section should include, where appropriate ... an informal introduction to the topic, without rigor, suitable for a general audience." In fact, the main part of these edits amounts to replace a sentence that exactly follows above recommendation by a much more WP:TECHNICAL definition. The remainder of the edits consist of adding technical details that, in any case are not relevant for the lead. IMO, they are also not relevant for the body, as they partly depend on the choice of a particular definition of an algebraic structure, and the same information appears already in the article with a more neutral point of view.

To Vfp15: Please, respect the guidelines of WP:BRD, that is try to reach a consensus by discussing here, and wait a consensus before editing this lead again. D.Lazard (talk) 17:02, 4 April 2017 (UTC)

The lead was incorrect in stating that a field was set with four operations. My initial edit of the lead was considered too technical by D. Lazard who reverted my edit instead of improving it. My subsequent edit took his criticism into account. Specifically I removed a technical term (algebraic structure) and simplied notation (1/${\displaystyle a}$ instead of ${\displaystyle a^{-1}}$). I made a few cosmetic and simplifying edits after that, which D. Lazard simply reverted. His behaviour is not, in my opinion, constructive nor in search of a reasonable consensus. Vincent (talk) 17:11, 4 April 2017 (UTC)
To D.Lazard: Please respect 3RR rule and have a look at the "Dispute Resolution" box in the arbitration requests page. Specifically "Assume good faith" and "Be open to compromise".Vincent (talk) 17:16, 4 April 2017 (UTC)
As Vfp15 has reinstalled his last edits without discussing here, I'll detail why they are not convenient:
1. "Operation" is unnecessarily technical for the first sentence
2. Saying that "addition, subtraction, multiplication, and division ... behave as they do when applied to rational and real numbers" is sufficient for describing, in the lead, the properties of the operations. More details, as given by Vfp15, may be confusing for the layman. These details appear in the body; repeating them in the lead is not useful. Moreover this may make less visible the description of the usage of fields.
D.Lazard (talk) 17:28, 4 April 2017 (UTC)
While I agreed that "algebraic structure" was perhaps too technical, "operation" is a term used in grade school. Hardly technical. As for being confusing to the layman, the first sentence is very simple and it is, moreover, correct. It is incorrect to say that there are four operations and imply they are distinct operations. Also the subsequent paragraphs in the lead intro mention scalars and vectors, which are much more technical terms than is the term operation. Vincent (talk) 17:47, 4 April 2017 (UTC)
Also, I do not appreciate your incorrect statement that I did NOT discuss the changes. I did most certainly discuss and in the spirit of compromise I integrated your comments into subsequent edits. You on the other hand reverted without discussion and began discussing only after reverting a third time. Again, I repeat that I am open to compromise. Consensus means seeing the other person's point of view, which I did and you did not. Vincent (talk) 18:07, 4 April 2017 (UTC)

The lead section currently contains the following sentences:

Two more operations associated with fields, subtraction and division, are simply the inverse operations of addition and multiplication. Like rational and real numbers, fields must have a neutral element for addition commonly called "0" (zero), and an identity element for multiplication commonly called "1" (one). Additionally, every element ${\displaystyle a}$ of a field must have an additive inverse commonly called the negative of ${\displaystyle a}$ and written ${\displaystyle -a}$; every element ${\displaystyle a}$ of a field except 0 also must have a multiplicative inverse commonly called the reciprocal of ${\displaystyle a}$ and written 1/${\displaystyle a}$.

In my opinion, this paragraph is highly suboptimal:

• "simply" -- unnecessary and unencyclopedic.
• "inverse operations" -- terminology which is unclear to the expected audience of this article.
• "neutral element" -- ditto.
• "commonly" -- weasel word (and also unnecessary, I personally have never seen an author not denoting these elements by 1 and 0)

What, on the other hand is entirely missing is an explanation, say, that the integers do not form a field. And it is also worth mentioning why (in a terminology that should be avoided in the lead) multiplicative inverses are necessary: to solve linear equations. Jakob.scholbach (talk) 20:20, 4 April 2017 (UTC) Jakob.scholbach (talk) 20:20, 4 April 2017 (UTC)

I have reverted the lead to the way it was in January, before the edit warring began. If the wording is to be debated, the article should be left in its prior state until consensus is reached.
The lead should be written in a way that can be understood by someone who does not know what a field is, and is striving to find out. The version I have removed failed in this. It began with the paragraph

In mathematics, a field is a set with two operations, commonly called addition and multiplication, which behave as they do when applied to rational and real numbers.

This is indeed easy to understand. A reader is likely to think "I see, that makes sense. It also includes the integers. I wonder why it doesn't mention those". Then he may go on to read the second paragraph, which contradicts the first. The reader is likely to give up at that point. Trying to learn a new concept from a source that contradicts itself is generally a waste of time.
I am in favour of any change that makes the lead of this article easier to understand. But Vfp15's change makes it harder to understand. Maproom (talk) 21:38, 4 April 2017 (UTC)
I disagree that it makes it harder to understand, but I do agree it might lead to a mistake because integers have two operations and yet are not a field. However, as it stands now it is wrong. There are groups (one operation with constraints) rings (two operations) and fields (two operations with more constraints). Saying four operations is just as confusing and incorrect. Further, fields are an algebraic concept, and so fields are very much about structure ie a set and something else that gives structure to that set. But then, why mention sets at all? Why not say "A field is like the rational and real numbers."? A set is also a technical term.
When I first I landed on this page my impression was "Wait a second, that's not right." Why? Because the definition is wrong. Even leaving aside equating a field with a set, only two operations are defined for fields, not four. Vincent (talk) 21:37, 4 April 2017 (UTC)

I really think this whole article has room above for improvement, perhaps prominently the lede. I am convinced that 'simply reverting' any change cannot achieve any evolution. However, I welcome the reversion to the status quo ante by Maproom in this warring environment, and I also agree to the estimation by Jakob.scholbach.

One might argue about defining fields via 2 'groups', or 4 'operations', or 2 operations with 'constraints', involving the notions of 'sets' or 'algebraic structure', mentioning 'rationals', 'reals', and 'integers', about hinting to different 'constants' and all the other important technicalities. I think this is all righteously content of this article, and some of them even might get their first alluding to within the lede, but this lede should start with 'innocent' terms, the layman associating meaningful interpretations with it. I disagree to 'algebraic structure' being too technical a term, and assume a meaningful association within an average reader, and therefore suggest:

In mathematics, a field is an algebraic structure, which is defined to abstractly model the behaviour of every-day-life numbers under the usual four arithmetic operations (addition, subtraction, multiplication, and division). These numbers are represented in the mathematical generalization by appropriate sets, and the arithmetic is modeled as mappings of elements of this set to (other) elements. Fields are thus a fundamental algebraic structure, which is widely dealt with in abstract algebra itself, number theory and many other areas of mathematics.


Perhaps this contains some crumbs to agree on, or even to improve on. Purgy (talk) 07:33, 5 April 2017 (UTC)

This is a so-so article with a very bad lead, but I will not waste further time on it. Life's too short to waste on edit wars with certain people. Cheers, and good luck!Vincent (talk) 13:30, 5 April 2017 (UTC)
I have invited people at WT:WPM to help with this article. The lead is not good, but the article itself is also in a poor shape. Let's all work together, rather than reverting ourselves to make this article decent! Jakob.scholbach (talk) 16:45, 5 April 2017 (UTC)
I think I know what a field is, but I can't follow Purgy's proposed definition, it's worse than Vfp15's. "These numbers are represented in the mathematical generalization by appropriate sets"? What is "in the mathematical generalization" doing there? It certainly doesn't make the definition any clearer. The same goes for "appropriate". So, "These numbers are represented by sets". Does this mean that each number (each field element) is represented by a set, like with Peano?
Ok, this is difficult. I admit that I couldn't do any better myself. The current lead may not be great, but it does explain, in terms I can understand, what a field is. Let's not replace it by something worse. Maproom (talk) 17:41, 5 April 2017 (UTC)
I'd like to mention that the suggestions, which you critisize, cover the notion of field beyond the scope and in more detail than the praised status quo does, so no big surprise that they are improvable.
• In my first sentence, roughly mirroring the actual one, I'd like to exchange "every-day-life" with a more idiomatic expression, not at my hands, but want to uphold avoiding to talk about the expert-only difference of reals and rationals.
• "These numbers" in my second sentence refers exactly to the "every-day-life" ones of the first sentence.
• The result of "the mathematical generalization" and formalization of the object described in the first sentence is exactly the notion of "field", currently under definition.
• The "appropriate sets" are intended to denote those sets that are capable to carry the required structures to make up a field. In particular, they may be the rationals, the reals, the complex, the quaternions (for skew fields), certain finite sets and their isomorphic pics. I'm in doubt that these are apt to show up in the lede, more concretely than just being "appropriate" ones.
• No, "these numbers" are not considered as in the von Neumann construction of naturals, but refer to the introductory vagueness of "every-day-life numbers", which are to be concretisized and generalized, too, to serve their math intentions.
Finally, I want to ask for help on the variants, which are around to define the notion of fields. Certainly, I do need cooperative help to achieve some top-level idiomatic text, but I consider it as not that hard to improve on the current one. Purgy (talk) 08:08, 6 April 2017 (UTC)

## What is wrong in the article

I agree with Jakob.scholbach that the whole article deserves to be revisited. Here, I will try to list what has to be changed. This is clearly my own opinion. Moreover I have certainly forgotten many points.

• The article appears to be written only for people that know the general theory of algebraic structures, and are more interested in this general theory than to the particular structure of fields. This is clear, at least, by the numerous occurrences of "algebraic structure" and "operation". IMO these should appear only in a specific section "relation with other algebraic structures", and I am not sure that this section is needed: this article is not the place for discussing the list of ring-like and group-like structures.
Could not agree more! Let's trim these occurrences to an absolute minimum, using footnotes where sensible for such technical digressions. Let's target the article (at least the easier parts) for an audience consisting of interested 15year old kids. They don't need weasel words like algebraic structure, but rather need gentle explanations of the purpose of the axioms, for example. Jakob.scholbach (talk) 18:27, 6 April 2017 (UTC)
• A section "Properties" is missing. Here are some of them that are either missing or difficult to find in the article:
• Subtraction and additive inverse need not to appear in the definition, only the constant ${\displaystyle -1}$ is needed, as ${\displaystyle -a=(-1)a}$ and ${\displaystyle a-b=a+(-1)b}$ (this is important in constructive mathematics, and used internally in many computer algebra systems.
I don't understand why people here are so much addicted to discussing the different versions of axioms. IMO, we should stick to the standard definition which is the one we currently have (+, \cdot, additive and multiplicative inverses). We should then gently explain how these axioms yield - and /, but any further description of equivalent axiom sets should occupy at most 5 lines, and it should appear in the article rather late. Definitely not before the examples! Jakob.scholbach (talk) 18:27, 6 April 2017 (UTC)
I believe I have badly expressed my point. I do not want to change the usual definition. What I am talking about is a property, which asserts that above identities are always true, and which allows (in computers) an easier use of associativity law and an easier regrouping of like terms. This should be explained among many other properties, not in the definition section. D.Lazard (talk) 19:33, 6 April 2017 (UTC)
I disagree, Characteristic (algebra) is IMO the right home for this notion. Here, we should briefly introduce the concept, refer back to the introductory example of F_4, and briefly discuss why this characteristic zero / non-zero is such an important dichotomy. Jakob.scholbach (talk) 18:27, 6 April 2017 (UTC)
• Every field homomorphism is injective. Thus thus the study of homomorphisms is replaced by the study of subfields and extensions
• Definition of the different kinds of extensions, structure of simple extensions, primitive element theorem, ...
• Galois extensions, fundamental theorem of Galois theory, ...
We do have part of that, but it is written in a way that is maximally annoying. Again, we need motivating examples, for example the Galois group of a quadratic extension should be related to the usual formula for solutions of degree 2 polynomials. Jakob.scholbach (talk) 18:27, 6 April 2017 (UTC)

Many of these have their own articles, but a user looking for Field theory is redirected here; an overview of field theory is thus needed. D.Lazard (talk) 14:45, 6 April 2017 (UTC)

Partly dissenting opinions:
• I think Abstract Algebra is dignified enough a math topic to justify a lede in this article to be written from its stance, and not from some arbitrarily selected (standard = usually = "in my education") POV. This is not to say that I prefer the notion of field informally introduced as a special division ring, but rather that I still valuate very high the intuitive introduction, as it is done now and as I proposed in my first sentence in a similar manner, which is also historically reasoned (Zahlen-Körper), considering a field as some closed entity to operate on in a very fundamental way. I think the currently accepted professional view on fields is succinctly traceable from there.
• While I agree to the need of gentle explanations of the purpose of the axioms, I oppose to calling algebraic structure a weasel word, I rather claim that interested teens associate a quite coherent notion with this, even when all the higher algebraic plunder is not yet available to them.
• I strongly believe that comparing different approaches to the notion of field allows for an eased understanding of the mentioned identical purpose, achieved by different sets of axioms. Selecting one such set as the standard or usual one is more weasely to me than the innocent use of algebraic structure in a non-expert meaning.
• In the light of the previous point I do not understand, why the hint to portraying the inverses as result of unary functions to ease the access to topological structures was removed.
I hope for argumentative conversation. Purgy (talk) 08:04, 7 April 2017 (UTC)

## Other definitions of a field

Emil Artin opens his 1944 Notre Dame lecture "Galois Theory" by defining a field this way:

A field is a set of elements in which a pair of operations called multiplication and addition is defined analogous to the operations of multiplication and addition in the real number system (which is itself an example of a field). In each field F there exist unique elements called 0 and 1 which, under the operations of addition and multiplication, behave with respect to all the other elements exactly as their correspondents in the real number system. In two respects, the analogy is not complete: 1) multiplication is not assumed to be commutative in every field, and 2) a field may have a finite number of elements.

This is a crystal clear definition where no technical terms are used save for "real number system" and "commutative". (I do not consider "operation", "multiplication or "addition" to be technical terms because it's grade school stuff.) It touches on all the essential elements of the definition of a field and avoids making false statements.

Wolfram World defines a field this way:

A field is any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.

Now this is shorter but it is certainly technical since it assumes the reader knows the difference between the mathematical structure called an "algebra" and the mathematical topic called "algebra".

On French Wikipedia, a field is defined this way:

En mathématiques, un corps commutatif est une des structures algébriques fondamentales de l'algèbre générale. C'est un ensemble muni de deux opérations binaires rendant possibles les additions, soustractions, multiplications et divisions. Plus précisément, un corps est un anneau commutatif dans lequel l'ensemble des éléments non nuls est un groupe commutatif pour la multiplication.

or (my translation)

In mathematics, a field is one of the fundamental algebraic structures of general algebra. It is a set provided with two binary operations making possible addition, subtraction, multiplication and division. More precisely, a field is a commutative ring in which the set's non zero elements is an abelian group for multiplication.

While this definition mentions all four familiar operations, in correctly states that only two operations are required for all four to be possible.

As it stands the lead is incorrect because it states there are four operations rather than two, and (more forgivably) it is incomplete because it does not include the idea of commutativity or of field axioms. Any of the three above quotes would be an improvement over the current lead. Vincent (talk) 15:23, 6 April 2017 (UTC)

Firstly, the lead does not intend to give a definition, but only "an informal introduction to the topic, without rigor, suitable for a general audience" (quotation of MOS:MATH). The place of a definition is in section "Definition". Secondly, for most things, including fields, there are several equivalent definitions (equivalent means that if an object satisfies any one of the definitions, it satisfies all of them). None of these equivalent definitions may be considered as being more or less correct than the other, and the choice is only a matter of taste, or sometimes of context. As the number of field operations appearing in a definition depends on the chosen definition, it is useless to search sources for fixing this number. Thirdly, the minimal number of needed operations may depend on the logical form that is accepted for the axioms: if you accept existential quantifiers, addition and subtraction are sufficient. If you do not accept them, two operations are also sufficient, but they cannot be addition and subtraction, they should be subtraction and division. If you want to include addition and subtraction in the primitive operations you need a unary operation (multiplicative inverse) and three nullary operations (constants), 0, 1 and –1. D.Lazard (talk) 17:23, 6 April 2017 (UTC)
I like the idea of opening our definition section with an outline of the definition, and the above quote of Artin sounds good for that purpose. Vincent, can you add it? (with a reference!)
Wolfram and French Wikipedia don't do a very good job in offering an accessible approximation to the actual definition (what does "commutative divison algebra" and "abelian group" mean for a layman? -- nothing). That said, let's work on the lead once the article is finished. Jakob.scholbach (talk) 20:17, 6 April 2017 (UTC)
I would be careful about such an addition. Artin's fields can be non-commutative(i.e., proper division rings), a terminology that is no longer very popular. The intent of that last sentence in the quote is good, but I don't think we should include it as it stands.--Bill Cherowitzo (talk) 21:38, 6 April 2017 (UTC)
I agree with Bill. IMO, Artin says essentially the same as the first sentence of the lead. The difference lies in commutativity (nowadays, fields are commutative), in the fact that fields may be finite, and in the fact that, in Artin's formulation, it is not clear that the integers are not a field. D.Lazard (talk) 08:56, 7 April 2017 (UTC)

## Recent edits

These recent edits by D.Lazard are, in my opinion and with all due respect, a big step back.

• removing the addition and multiplication table for F_4 is not a good move
• what is now completely missing is an explanation of the definition in light of the examples; these two sections are now essentially disconnected, which they should not!
• a distinction between examples and "fundamental examples" is artificial; also the "fundamental examples" are poorly formatted (by WP:MOS, avoid bold fonts)
• the section on fundamental examples mentions many notions which at this point are not at all relevant and also not accessible to our audience (remember -- a 15 year old kid!), for example "subring", "up to a unique isomorphism"; similarly for the entry about the reals and the complex numbers.
• why do subfields appear this early?
• [I could go on!]

I am almost on the edge of immediately reverting these edits, since I am convinced it is much more time-consuming selecting the good parts of these (good-faith!) edits than continuing with the version we had before. D.Lazard, can you quickly comment on these edits? Or anyone else? Jakob.scholbach (talk) 01:24, 9 April 2017 (UTC)

Even when not being satisfied with the result of several edits by D.Lazard, I cannot easily agree to reverting to the previous version would make the improvement easier. I really think that the overall structure of the article has straightened, and that it is easier to discuss and improve on D.Lazard's version than on the previous one, provided D.Lazard admits editing of his texts and structures. Furthermore, I consider it to be easier to work from the current version, and judge from the immediate effects.
Just for convenience in the above sequence:
• I consider a solid example of a finite field with operation tables to be very illustrative, even when the F_4 example might theoretically belong rather to extensions. Perhaps a prime bigger than 2 (F_5?) might fit.
• I agree.
• The "fundamental examples" are more or less the exhaustive hierarchy of a certain class of fields, more apt as paradigmatic structures (isomorphism!) than as didactic examples.
• Yes, this section is freighted with new notions, instead of shining light on defined ones. E.g., the introduction of the notion prime field is imho misplaced. Additionally, I do not believe that a binary field adds much to understanding of "computing with bits".::*
• I'm not sure if subfields are better at home with extensions.
• [I wont go on, either]
Belonging to anyone else, and giggling about the introduction if i² = -1. Purgy (talk) 07:32, 9 April 2017 (UTC)
I do not well understand above criticism. May I recall that Wikipedia is not a textbook, and that our audience is far to be restricted to 15 year old kids. As fields are used not only in every parts of mathematics, but also in many other scientific and engineering areas, this article must also be written for these people. Let us consider the above points:
• The operation tables of F4 appear in Finite field. As operation tables of finite fields are almost never used, except as classroom exercises, I see no reason to reproduce them in this article. Thus, if this example should appear somewhere in this article, this should be done with the details (and the tables) replace by a link.
• I do not see any explanation that I would have removed. I have removed only a partial proof that the rational numbers form a field, which explains nothing. On the other hand I have introduced the main explanation, which was lacking, namely that the concept of field has been introduced for abstracting and unifying properties that had been encountered in various situations.
• I agree to remove bold fonts, and I have done this. "Fundamental" has to be taken here in its proper meaning, namely that field theory has been built on these examples. Nevertheless, I would agree to call the section "Basic example".
• The examples In the section "Fundamental examples" has been written in a way that allows an easy skip for a reader who has never heard of some examples. Let me recall that our audience comprises also engineers who often know better the complexes and the reals than rational numbers, and computer scientists that are more accustomed to the binary field rather than any other field in the list. When writing these examples I have tried to explain why they are important, and I have intensionally left the details for the linked articles. However, because these terms are essential, I have explained "complete" and "algebraically closed".
• "Subfield" appears so early because the concept is absolutely fundamental, because, in the previous version, it appears before being used, and because it is needed for including in the future "Properties" section the fact that every field contains a smallest subfield, which is a prime field.
By the way, I plan to add the following sections, between the "Fundamental examples" and "History". The place of "History" may be discusses later, and most existing sections will be merged in these new sections.
• Properties (including characteristic, prime subfield, the fact that a field is an integral domain that has only one proper ideal, ...
• Constructing fields (most examples should be moved in the subsections)
• Quotient by a maximal ideal
• Field of fractions, including, as examples or subsection, field of rational fractions (fractions of polynomials), Laurent series, ...
• Simple algebraic extensions
• Completion with respect to a valuation maybe here, maybe nearer to the bottom
• Field extensions: the different classes of extensions (finite, algebraic, transcendental, separable, inseparable and purely inseparable) (only a summary, details left for Field extension. Galois theory should also appear here probably as a subsection
D.Lazard (talk) 10:30, 9 April 2017 (UTC)
A few comments. To me, overall, recent edits seem a big improvement. About a multiplication table. I have always been puzzled by them. Yes, one can draw a table but for what end (I don't think it's even useful as a class room exercise; the students learning fields are more intelligent than first graders.) The field of real numbers (or complex numbers) is elementary, intuitive and serves much better as a basic example. To Lazard's list above, I would add a topological field; it probably not a bad idea to briefly mention GL(F) when F is a topological field. I suggest we also discuss a vector space briefly; without this background, a field extension makes sense but is quite trivial. Also, the article makes no mention of model theory; this is not good (though I cannot do anything about it myself.) Some useful references can be found in http://alpha.math.uga.edu/~pete/expositions2012.html -- Taku (talk) 20:41, 9 April 2017 (UTC)
[edit conflict] I continue to strongly disagree with your explanations. Let's include an engineer, next to our 15 year old, into our target audience. However, as I will try to make clear below, this is not in any way improving the quality of your edits.
Writing explanations that can easily be skipped is not a sensible idea, IMO. What is the point of mentioning that Q contains Z as a "subring": neither the 15 year old, nor the engineer understand this sentence. You convey an amount of zero information to those people with this sentence. While they can easily arrive at an understanding of this, your structure prevents this from happening.
Any other of your fundamental examples is just written in a way which prevents learning and understanding of points our readers don't already know. This includes your engineer. (I mean this as gentle as I can, but here it is hard to say it differently). Good mathematical writing, IMO, has a constant slope so to say. I.e., don't jump in difficulty, abstraction, or complexity, but rather try to start with basics and then gradually increase the necessary abstractions etc. Your explanation just jumps from zero to a high level, in a way which is unhelpful and also unnecessary. We do have a certain space for explanatory material here, and this is the article to give these explanations. Not being a textbook does not mean we should, as a general rule, avoid introducing concepts and giving explanations.
Removing explanatory material such as the multiplication table is greatly diminishing the possibility that someone would understand something new here, again particularly when we think about an engineer / computer scientist (who might know about xor / and; and might have an aha moment when this embedds into something slightly bigger). You ironically also have included into the definition of F_2 a summary of that table in prose, which is way harder to understand than such a table.
You did remove fundamental pieces of information, such as the definition of C. What you currently offer is a theorem about C which however is not conveying any meaning to someone who does not know C already.
I am also not convinced that your planned new structure of the article is a good idea. Take function fields, for example. While it is true that these can be obtained as finite extensions of F(x_1, ..., x_n), which in its turn is obtained from F by a transcendental extension, this way of constructing the field is not what makes the field important. What makes it important is the fact that it represents functions (whence the name). Forcing all the examples of fields we (have to) discuss into the corsett of "how do we get them" is not helpful at all. Jakob.scholbach (talk) 20:43, 9 April 2017 (UTC)
I have not removed any information (except tables for F4). I have replaced the (very sketched) definitions of Q, R and C by links to the article that give detailed definitions, explanations and proofs that they are fields. As the links appear at the very beginning of the paragraph, it is very easy for the reader to go there if he needs details and learn what are these fields. We have not to choose in his place. If he does not know, or knows vaguely what is C, it is his choice of learning it from the linked article, or being satisfied to learn that C is important and why, or to completely ignore the paragraph. Again, we have not to chose for him, nor to make hypotheses on his background.
About tables for F4. Except for verifying that F4 is a field multiplication tables are not useful at all, and this is why nobody (except some teachers) use them. Even for verifying that F4 is a field the table is not very useful: the verification of associativity, for example, is not simple and very boring. If really you want tables, go to Finite field#Field with four elements, where tables are more informative.
I am not sure of what has to be done for the fields of functions of algebraic varieties (I do not know any other function field). They could appear as a further example of a field of fractions (field of fractions of the ring of regular functions). They could also be the subject of a section "Fields in algebraic geometry". In any case, the present section "Field of functions" is a mess, where the reader normally understand that the continuous functions on a rectangle form a field! D.Lazard (talk) 17:59, 10 April 2017 (UTC)
Yes, the section on function fields can use some improvements! Nonetheless, as I said forcing this and other examples under the headings "how is this field constructed" is not helpful, IMO. Let me illustrate this argument applies with some of the other fields currently in the examples section: the field of constructible numbers can be characterized purely algebraically, however this is not the initial reason why we look at it. Number fields are defined being finite extensions of Q (and the concept of a field extension is obviously crucial to this article as well), but our interest in them stems not so much from this fact but rather from the development of number theory (for example Kummer's approach to Fermat's last theorem). Similarly, even though Q_p is defined as a completion of Q, this is not the primary reason we look at it: we look at it since it tells us something about congruences of numbers. We don't primarily study complex numbers since they satisfy the conditions you mention in "Fundamental examples", but use it since it is just so useful in mathematics, physics and engineering. A good article about fields should, IMO, illustrate why these fields exist as an object of study. While all of this is possible by ordering all the examples in headings like "finite extensions", "completions", "residue fields" etc., this would make a rather arid article.
Asserting that you did not remove things is simply wrong, you did remove the definition of C, which is a fundamental piece of information when we talk about the complex numbers in an article like this. Please kindly look at the diff link I pasted above and insert it back, otherwise I or others have to do it.
The fact that such group tables are used in "classroom exercises" indicates their ability to convey understanding, which is our goal. Again, I will reinsert the example you deleted since it shows some basic features of fields and allows us to later refer to this example when explaining, say, field extensions, the characteristic and other features.
Your comment "We have not to choose in his place." strikes me as strange -- after all you(r edits) do choose for the reader: before we had a (brief) explanation of the key concepts such as an indication of the proof that Q is a field or the definition of C (and a link to the subarticles). After your edits (in the fundamental examples section) we are only left with three hardly illuminating characterisations of these fields which don't foster understanding the concept of a field. I am not against mentioning these theorems later in the article, but as a first invitation to the concept of a field, they are simply useless. Jakob.scholbach (talk) 03:11, 11 April 2017 (UTC)

## Sectioning

[These recent edits https://en.wikipedia.org/w/index.php?title=Field_%28mathematics%29&type=revision&diff=776156088&oldid=776124335] changed the section ordering: we now have a section "Fields with augmented structure". I personally have never come across this terminologya. Also, what is more problematic, global fields and field of functions are now listed under topological fields, which is misleading: while we could endow a global field with a topology resulting from a chosen valuation, the topology depends strongly on this choice; to an extent which makes the relation "global field -- topological field" useless (and also, as far as I am aware, non-existant in the literature). I am firmly convinced the structure we had before was more appropriate.

Also, in a similar vein, we now have "Alternative definition" right at the top. I don't have a 100% strong opinion about this, but I think it is a bit out of place there. These remarks fit, IMO, more smoothly later on when we have discussed, say, the additive and multiplicative group. (As a rule of thumb, when moving sections, it is good to watch out for internal "links" such as "As was mentioned above,", which in the present case is now broken.) Jakob.scholbach (talk) 15:41, 19 April 2017 (UTC)

In my edit I tried to resolve the two heaps "Examples" and "Generalizations and related notions" of the previous version by organizing their subsections in a more encyclopedic structure, thereby possibly sacrificing some pedagogical advantages. From this POV I put the variants of defining a field under under "Definitions". I think, the section on groups could be moved there, also.
"Finite fields" seemed to me sufficiently interesting to justify a section of their own. Together with the "History" section this makes up a first overview of fields in general, whereas the rest digs somewhat deeper, first with putting fields in a bigger picture ("Related algebraic structures"), next showing how to make up new fields ("Constructing fields") from their relatives, then augmenting (adding to, enriching, ...) their "structure" and ending in "Galois theory" and "Application". This appears to me as a quite nice structure of the article.
I did not intend to refer to some technical term with "augmented structure", but just wanted to point to -often readily available- structures, which may be on top of the basic field properties, as there is an "order" or a "topology", a "derivation", ...
Since I did not expect that my suggestion would be accepted wholesale, and I wanted to make explicit that I did not change anything in the verbiage, I did not care about wrong forward/backward referrals, and other clerical details like changing the "Fundamental examples" or introducing a subheading like "Classic definition".
As regards the subsections "Function fields" and "Global fields", I do not see them more misplaced now, than they were before in the unstructured heap of "Examples", even when their -necessarily referred to- topology may be arbitrarily selectable, and I emphasize again that I did not change the verbiage of the whole article, so I did not deteriorate the subsections, and any upgrade is feasible.
For the above itemization I am honestly committed to the new structure being superior to the previous one, obviously not denying potential for further improvement.
Hopefully, I addressed your concerns. Regards, Purgy (talk) 07:41, 20 April 2017 (UTC)
OK, I am still not convinced, but based on your edits I suggest the following structure. These are also somewhat closer to the above ideas of D.Lazard, I believe.
1. Definition and fundamental examples (Definition, Q, R, C, constructible numbers, F_4)
2. Elementary properties (Elementary consequences of the definition, Additive and multiplicative group, alternative definition, char., prime fields)
3. Finite fields [I am not entirely sure they deserve such a top-level place]
4. History [could be moved somewhere else, possibly]
5. Constructing fields (Rings vs. fields, extensions, closure, ultraproducts)
6. Fields with additional structure (ordered fields, topological fields [including local fields], differential fields, exponential fields [I believe this last subsection will be massively trimmed, it testifies its own uselesness])
7. Galois theory
8. Applications (linear algebra, commutative algebra, number theory [including global fields], geometry [including function fields])
9. Related notions (skew fields, Fields)
I am still contemplating where to put material related to the model theory of fields, such as the Lefschetz principle and other related facts. Jakob.scholbach (talk) 18:47, 20 April 2017 (UTC)
I stand by my comment that global fields and function fields are completely misplaced in the section on topological fields. The topology induced by a given place is highly dependent on this choice: Q endowed with the p-adic or the usual absolute value makes two entirely different (i.e., non-isomorphic) topological fields. The point that we can choose a topology on Q or k(t) is not indicating that we should view (nor present) them as topological fields. Compare this to the obvious remark that we can choose a trivial norm (hence topology) on any field. Jakob.scholbach (talk) 18:52, 20 April 2017 (UTC)
Thank you for considering my thoughts to this extent. Just to comment on your and my own reservations:
1. I slightly hesitated while putting "constructibles" into the "Examples", but did not find a better place. This section might be considered somewhat bloated.
2. Besides continuing improvement I see currently no particular problem.
3. I share your reservations from the purely abstract POV, but I think that their ubiquity in applications, and their juxtaposition to the everyday continuity of the reals makes them a sufficiently interesting topic, especially for newbies, and thus worth an extra section, including the OP-table.
4. Yes, "History" is certainly movable. However, personally I like these first 4 sections gruoped together, making up an easily graspable, reasonably complete first look, without too much perspective on any abyss.
5. I take from your comment that you do away with the section "Related structures", which I intentionally kept at this place to introduce the rings with this nice table (not really of fundamental importance, but aerating the setting), having them ready for constructing fields. Putting into this section your model theory and Lefschetz principle and my K-algebra appears misplaced to me, but having rings at hand befor delving into "Construction" might be useful, still. Merging just the table into "Construction"?
6. I think, meanwhile I get a feeling for your strong opposition on simply subsuming global fields and function fields under topological fields. May I further to my excuse, that I considered the current text on these topics as too small as to justify a more soloistic role.
7. OK, "Galois theory": I am absolutely clueless to what volume this should go.
8. Bolstering "Application" in this way is fine to me. Could this be a place also to put an end for your contemplations? :)
9. May I doubt the necessity of a section "Related notions"? I think skew fields and Fields could simply find a suiting place under "Additional Structures", perhaps retitled to "Modified structures" (I used augmented exactly to hide this nuanced meanings).
BTW, I am not cognizant of D.Lazards intentions and motives. Best, Purgy (talk) 10:37, 22 April 2017 (UTC)

I don't think skew fields, Fields, the field with one element (and other related notions we might include) should live under the same roof as augmented fields. These two deviations go clearly in the opposite direction: in one case we consider field with more structure, in the other case we consider (in a vague sense) fields with less (axioms etc.). I have placed the related notions at the end according to my proposal. Jakob.scholbach (talk) 04:19, 30 April 2017 (UTC)

From my perception of structure all the "related objects" should be dealt with under one roof, whatever this roof is called. I am not really fond of the current classification of nimbers, surreal numbers, Fields, and skew fields in the current hierarchy. Purgy (talk) 15:12, 30 April 2017 (UTC)
I agree "related notions" is not a terrible good heading (do you have a better one?), but in many articles there is just a number of notions which are related to the article, but not necessarily in a way that allows for sensibly grouping these neighboring notions. I don't see how to do otherwise. In any case, Fields do not belong to fields with additional / augmented / ... structure since they are not fields. Similarly, skew fields are out of place in a section "Constructing fields [from commutative rings]". Jakob.scholbach (talk) 02:15, 1 May 2017 (UTC)
Sorry, for the moment: no. The only idea that crossed my mind was to put a collection of constructs into "See also", since pointing to e.g. "Linear algebra" as "main" below a subsection header is also not really my first choice. I think I'll take a creative break on this question, and watch the development of contributions on "Related objects". Perhaps I may then submit an idea I consider worth mentioning it. Purgy (talk) 12:08, 1 May 2017 (UTC)

## Exponential Fields

I see that Jakob.scholbach has deleted the paragraph starting "One does not" from the "Exponential fields" section. I think this is a shame. It was a well-written and informative paragraph, which answered a question likely to be asked by any intelligent reader seeking to understand fields. Maproom (talk) 07:46, 23 April 2017 (UTC)

I agree it was relatively well-written, but I still think this paragraph was out of place in an article like this. Maybe, in a textbook, there is space to explaining why certain notions don't make sense (or are uninteresting). A WP article, however, has to focus on summarizing knowledge, not preventing readers thinking about uninteresting / irrelevant notions. For the same reason, we don't include, say, typical misconceptions in understanding a concept.
If we are to mention this fact at all (which I think we shouldn't), this deserves about one line in this article, which has so much more to say. Jakob.scholbach (talk) 01:37, 24 April 2017 (UTC)
I do not dare to judge the importance, not even their frequency in the relevant literature, of "exponential fields", and I do understand the reservations to include caveats and "not-interesting"-facts in an encyclopedia, but if one accepts "Exponential fields" in this article, then there should be included some considerations about how exponentiation is dealt with. Supplementing such terse information with "why not an other way round" is, imho, very desirable from a readers perspective. I tried to shorten the original text and throw this into discussion below:
Note that a binary exponential operation ${\displaystyle \scriptstyle a^{b}}$ is neither associative, nor commutative, nor, unlike addition and multiplication, has it a unique inverse operation(e.g., ${\displaystyle \scriptstyle \pm 2}$ are both square roots of 4), nor is it even defined for many pairs (e.g., ${\displaystyle \scriptstyle (-1)^{1/2}\;=\;\pm {\sqrt {-1}}}$) in e.g., the field of rational numbers. Adding exponentiation along the familiar path, from addition over multiplication to exponentiation, each induced by iteration of the last, has also not proven fruitful; instead, a group homomorphism from the additive group to the multiplicative group is assumed as a third, unary operation ${\displaystyle ({\scriptstyle \exp(a+b)\;=\;\exp(a)\cdot \exp(b)})}$ in an exponential field.

I do not at all strongly adhere to any of the options (leaving as is, restoring from history, improving my suggestion, ...). Purgy (talk) 08:58, 24 April 2017 (UTC)
I like the proposed paragraph above – though I would consider removing the   ${\displaystyle \;=\;\pm {\sqrt {-1}}}$   as more confusing than helpful. Maproom (talk) 09:29, 24 April 2017 (UTC)
IMO, both versions of the section are confusing, as not mentioning the widely used concept of exponentiation and logarithm in finite fields. In fact, in a finite field with q elements, there is an exponentiation with exponents belonging to ${\displaystyle {\mathbb {Z}}/(q-1){\mathbb {Z}}.}$ Because of the wide use of this exponentiation in everyday cryptography and in number theory, one may guess that most users searching for exponentiation in fields are looking for this discrete exponentiation. Thus, I suggest a section "Exponentiation in fields" where these two different exponentiations are presented (in two subsections?) in a way that avoids confusing them. D.Lazard (talk) 11:19, 24 April 2017 (UTC)
I agree it is a good idea to (briefly) explain discrete logarithms (which includes a brief explanation of a discrete exponentiation, i.e., a^b, where b is in Z). I suggest to put this in the applications section.
However, this is indeed an entirely different question whether we should include 5 lines of digression why a notion is not useful. If we were to include such things, we also have to include the (quite common, among students) misconception that F_q is, as a _field_, F_p^n. I think we can easily compile a list of "10 frequently errors related to fields". Yet, we don't include these here, and rightfully so. To the article about fields, the section on exponential fields is, IMO, quite tangential. To this, tangential, subtopic of exponential fields, the matter of (wrongly) tweaking the definition is even more tangential.
Just think of what we can do with the space occupied by the paragraph I deleted (or the one suggested by Purgy). We can do offer much more relevant information here. Let's do so! Jakob.scholbach (talk) 14:21, 24 April 2017 (UTC)
I do not want to be offensive with my last edit on this section, but I do think that just mentioning "some homomorphism" is not sufficient for a section on fields with exponential. Therefore I added the functional equation and a link. I also reestablished the subsection dealing with discrete exponents to take care of D.Lazards remark on where one would search for this, and, following Jakob.scholbach, I omitting the reasons for not following an obvious route. I tried to shorten the text as much as is possible to me, ridding the logarithm and pointing to Cryptography, even when I do not agree that the place, occupied by a paragraph in an online encyclopedia, is urgently to be saved for "more relevant" content. Maybe, relevance is in the eye of the reader. Purgy (talk) 14:58, 30 April 2017 (UTC)
No offense taken. Anyway, I don't think your edits on this section work well. Including a definition of a group homomorphism here is fine IMO, even though I think a reader that has read to this point is probably aware of the meaning, but let's leave it if you prefer. However, I don't think portraying exp as a unary operation is sensible (nor is this done in typical references, is it?), the point being that exp : F \to F^x is a map between different sets, in which case we don't speak of an operation, as far as I know. Moreover, negative exponents are nowhere needed in this article. Finally, the only point in introducing (--> "relevance") all this is the fact that discrete exponentiation is easy (i.e., quick), whereas its inverse is (in the right groups) hard. Abstract analogies between the definition of na and a^n are nearly that important than this. Since the article is getting long, and will include more material, I think adding these side-remarks is not helpful and distracting the reader's attention. Jakob.scholbach (talk) 02:23, 1 May 2017 (UTC)
May I, please, object to qualifying "exp() as map from F to F establishes a unary function on F" as "not sensible". If one aproaches the notion of fields from an algebraic view point, it is, imho, quite natural to look for "operations" on the set F, which carries the algebraic structure of the field F. The restriction to get from either codomaim F or F^x=F\{0} to the factual range of the map exp() just differs by {0}, so I do not understand why the reference to the sets, bearing the group structures, would hamper establishing exp() as one of the unary functions from F to F, which contribute to the algebraic structure of the exponential field.
I also hold against "Discrete exponentiation" being only relevant for its highly discrepant computational complexity in calculating some inverses in selected circumstances. Besides modelling a first approach to multiplication/exponentiation, I've seen this a lot in dealing with groups, and so I think that this construction is worth its minimal effort of introduction. Savings in the definition for omitting the negatives (zero is required) are negligible. Finally, I'm meanwhile convinced that searching for this keyword should find a reasonably closed (=beautiful) treatment, nota bene in some homonymous, if not analoguous environment, and that "space" is not one of the biggest concerns in an online encyclopedia ("structure" is!). :) Purgy (talk) 13:47, 1 May 2017 (UTC)
OK, here is a suggestion: try and look up 5-10 standard references about exponential fields and see how many of them present it being a group homomorphism (or equivalently using the formula you added), and how many on the other hand present it as a unary operation. If we have a rough equilibrium, let's include both versions. If not (what I strongly suspect), let's keep the approach I suggested.
If you discard my arguments or think that the article is really better off the way you prefer, just go on. I repeat and retreat to my stated earlier position of "I do not at all strongly adhere to any of the options". Here's to the best of the article. Purgy (talk) 05:33, 2 May 2017 (UTC)

## Ready to be a good article?

I believe this article is getting close to being a good article. I am planning on doing some work on it, but I would like to gather some feedback about what should be done before nominating it. I am looking forward to your opinions! Jakob.scholbach (talk) 01:44, 7 May 2017 (UTC)

Just for not leaving this uncommented, my 1/2 cent: I do not care a dime if some article is a "good" one in the sense of WP reviewers or not. This article recently gained certainly a lot of relevant, deep and even interesting content, it is has been brought to a remarkably unified styling (note also the efforts on layouting), but, nevertheless and without any weight and importance, it lost a lot of pleasure in reading to myself, and also in my estimate of the pleasure for an average visitor to this page, who is not fascinated by an, assumed necessary, encyclopedic terseness. For this I cannot recommend to read this article to anyone, who is not in the, almost perfect, know.
This article, imho, achieved a level, where judging it by the wisdom of the crowd(s) is simply ridiculous. Purgy (talk) 10:16, 26 May 2017 (UTC)
Yes, we have two competing goals: conveying understanding to our readers and keeping the exposition concise. Can you name some sections which you think should be more detailed / more accessible to readers with less mathematical background? In recent edits, I have tried to convey more motivation (which maybe is what you call "pleasure"?). More along those lines can certainly be done. It is hard or actually impossible for a reader to understand all what is written here by just reading this article. I don't know a way of avoiding this without either omitting much of what we know about fields, or without being extremely shallow. Jakob.scholbach (talk) 14:16, 26 May 2017 (UTC)
A WP editor, I believe it was CBM (?), roughly once said that WP is actually many encyclopedias. In theory, we could have an article Introduction to fields, Elementary properties of fields, ..., Cutting edge theory of fields (or other more meaningful titles). In some cases, such as General relativity vs. Introduction to general relativity, editors have taken this approach. I don't know if a similar thing could / should be done with fields: they just appear almost everywhere in mathematics in some form, so part of the difficulty of reading (and writing!) this article results from the fact that fields are so ubiquitous and studied to such a deep extent. Jakob.scholbach (talk) 14:24, 26 May 2017 (UTC)
A subsection of "Constructing fields", called "Completion with respect to an absolute value" is lacking. In fact ${\displaystyle {\mathbb {R}},}$ ${\displaystyle {\mathbb {Q}}_{p},}$ and ${\displaystyle F((x))}$ are all constructed by this process, and this deserves to be mentioned.
Some sections, such as "Fields (with capital F)", "Ultraproduct", "Model theory of fields", "K-theory" are very technical and useful only for specialists of a single domain of mathematics. Thus, they could advantageously be replaced by a single entry in the "See also" section. For the same reason, "The absolute Galois group" could be reduced to a single sentence in the section on Galois theory. D.Lazard (talk) 17:06, 26 May 2017 (UTC)
Thanks for your feedback! We do mention completion in the topological fields section, but I will carve it out a bit more prominently. (It could be made into a subsection of "constructing field", but since it is confined to metric fields, I believe the current placement also makes sense.)
I agree somewhat about Fields (this part was already here before I started working on the article). I like the example of a Field with nimbers (some games) though. I will think about it. I also see now that it makes sense to merge the ultraproduct section with the model theory section. However, removing any of these is not a good idea, IMO. Re Model theory: The Lefschetz principle is just such a common thing. I believe it deserves to be known by any mathematician, and is really a statement about fields. Re Galois group: The study of representations of Gal(F) is a huge topic. Much of arithmetic geometry touches this in one way or another. Re K-theory: Voevodsky's fields medal was based on proving the Milnor conjecture. Even though the statement itself is highly special, I think describing the quest of meaningful invariants of fields belongs here. I will think about making it more welcoming though. Jakob.scholbach (talk) 19:57, 26 May 2017 (UTC)

## Citations needed

Currently, the article reads "Fields can also be defined in different, but equivalent ways. One can alternatively define a field by four binary operations (add, subtract, multiply, divide), and their required properties."

It is obvious that one can define a field in this way. However, I have been unable to find a single reference which does it in this way. Does anyone know a reference? If not, I would suggest deleting the sentence. We do explain how to define subtraction from addition a few lines above. In my opinion this is enough, but if we find a (reasonably credible) reference suggesting that definition, I am fine with keeping it. Jakob.scholbach (talk) 02:35, 26 May 2017 (UTC)

In a similar vein, we read "Omitting the requirement 1 ≠ 0, would imply that the trivial ring had no multiplicative group, which is, according to definition, based on F\ {0}, since a group must have at least one element." Is there a reference for this statement or some other motivation why the trivial field (in which 0=1) is excluded? Again, I am leaning towards removing this, unless we find a reasonable reference expressing this point of view. Jakob.scholbach (talk) 02:35, 26 May 2017 (UTC)

Since the lede introduces the notion of a field via the four elementary arithmetic operations on reals (yes, really reals!), and it is not foreseeable that elementary math education will shift to the current view on defining operations, I think the possibility of doing it the elementary way should be mentioned, and be it by ignoring all rules of reliable sourcing. Perhaps one could remark that the math literature and real math at all is not very concerned about "subtraction" and (long) "division".
It may well be that I overlook some abstract importance of an empty group (I admit to be rather ignorant on abstract nonsense), but taking away the 0 from F(={0}) leaves F\{0} empty, and to me this is without multiplicative group, necessary to have a field.
Again, this is just because you mention it. Purgy (talk) 10:54, 26 May 2017 (UTC)

## Math notation

As a final step before a good article nomination (unless someone objects?), it would be good to streamline the mathematical notation. We partly have F, in other parts F, and (unavoidably for some formulas) ${\displaystyle F}$. I think it is good to have a uniform notation throughout the article. Are there any pros/cons? I personally prefer the F markup, but I am also fine with using the F markup (I just find it annoying to type). Jakob.scholbach (talk) 02:45, 26 May 2017 (UTC)

For my personal well being I do not require the least that the WP:MOS holds. My personal favourite is the [itex]-environment throughout. Sadly, it looks odd within inline rendering, but I do hate the differences for various glyphs in different fonts (especially for "a"). I think WP does not deserve all the effort of math editors on layout for them constantly ignoring their requirements. Less than a 1/10 of a cent. Purgy (talk) 11:11, 26 May 2017 (UTC)
I prefer F, for having a similar appearance in html and latex rendering. Several past discussions at Wikipedia talk:WikiProject Mathematics show that there is a consensus among mathematics editors for preferring {{math}} for inline mathematics, when possible. However there is also a consensus for not changing systematically the articles where the notation is coherent.
Passing from F to F is easy and rather fast: It suffices to select the formula to change and click on the button "Math and logic/{{math|}}" (care must be taken for formulas containing =).
By the way, the article does not respect MOS:MATH#Roman versus italic: the imaginary unit must be in italic. D.Lazard (talk) 11:14, 26 May 2017 (UTC)
OK--I was not aware there is a consensus about using F, but let's convert it to this format throughout then. Thanks also for fixing the non-italic i. Jakob.scholbach (talk) 14:04, 26 May 2017 (UTC)

## Small remark(s)

The lead: "The relation of two fields is expressed by the notion of a field extension" – rather, "a relation"? Boris Tsirelson (talk) 21:16, 30 May 2017 (UTC)

Sect. "Definition": "existence of additive inverses ... for all elements ..., and of a multiplicative inverse ... for every nonzero element ..." – could it seem that multiplicative inverse is unique, but additive inverse is not? Boris Tsirelson (talk) 21:24, 30 May 2017 (UTC)

Sect. "Subfields and prime fields": "If the characteristic of F is p (a prime number), the prime fields is isomorphic" — fields are? or field is?

Also, the Frobenius map is said to be "compatible" with addition and multiplication several lines before homomorphisms appear, and (strangely?) is not said to be a homomorphism. Boris Tsirelson (talk) 21:47, 30 May 2017 (UTC)

Thank you! Except for the first, all done. About "The relation": in my mind, if two fields are related at all, they are subfields / field extensions of one another. No? This is what I wanted to convey here, but if you think "A relation" conveys this better, go ahead and simply change it yourself. Jakob.scholbach (talk) 02:11, 31 May 2017 (UTC)

Sect. "History": "Lagrange observed that the function x1 + ωx2 + ω2x3 for ω being a third root of unity and x1, x2, and x3, only takes two values" – I fail to parse it. What takes only two values, the function? or x_1,x_2,x_3? On which set is the function defined? Boris Tsirelson (talk) 10:21, 31 May 2017 (UTC)

Thanks again. I am traveling, so cannot fix it right now, but it is supposed to say "takes only two values for the six permutations of the x_1, x_2, x_3". Jakob.scholbach (talk) 16:23, 31 May 2017 (UTC)

The same section: "what eventually became the concepts of fields and the concept of groups" – "the concepts of fields and of groups"? "the concept of fields and the concept of groups"? Boris Tsirelson (talk) 18:21, 31 May 2017 (UTC)

Sect. "Constructing fields": "to find a field (related to R), where all elements become invertible" – hmmm... rather vague... is it possible to make such umbrella for these two methods? Boris Tsirelson (talk) 18:33, 31 May 2017 (UTC)

"the field of fractions of the ring ... of power series" – really, formal power series. Boris Tsirelson (talk) 18:47, 31 May 2017 (UTC)

"${\displaystyle {\overline {a+bX}}\mapsto a+ib}$   – is it clear what is denoted by the \overline ? Boris Tsirelson (talk) 19:07, 31 May 2017 (UTC)

"Algebraic extention": "These are roots (or zeros) of polynomials" – the link should be to Root of a polynomial or Zero of a function, not Root. Boris Tsirelson (talk) 15:01, 1 June 2017 (UTC)

"Transcendence bases": "if F is an algebraic extension of E(S)" – is it clear what is E(S)? Boris Tsirelson (talk) 16:45, 1 June 2017 (UTC)

"Closure operations": "roughly speaking, F is not too big" – is it clear what is F? Ah, I see, this is said later; better late than never... Boris Tsirelson (talk) 16:51, 1 June 2017 (UTC)

"Ordered fields": "...every ..., which should exist, exist effectively" – I'd say "do exist", since there is no claim like "effectively computable" here. Also, shouldn't it be either "every ... exists" or "all ... exist"? Also, is it clear, what is "Cauchy sequence" in this context? Boris Tsirelson (talk) 18:45, 1 June 2017 (UTC)

All done. Thanks for your careful screening! Jakob.scholbach (talk) 21:03, 1 June 2017 (UTC)
Ridiculously, I am still confused with the cubic equation in the history section. May I take for example the three roots 1, 0, 0 of the equation ${\displaystyle x^{3}-x^{2}=0}$? Then the expression x1 + ωx2 + ω2x3 turns into 1 + 0 ω + 0 ω2 = 1, or, permuting the zeros, 0 + 1 ω + 0 ω2 = ω, or 0 + 0 ω + 1 ω2 = ω2; three values? Boris Tsirelson (talk) 08:30, 2 June 2017 (UTC)
Your objection sounds right. Bourbaki's "Elements of the history of mathematics", which is where this fact is taken from, is quite terse here. I will look into other sources how to correct the claim. Incidentally, our article cubic function has a pretty detailed write-up of Lagrange's method. Jakob.scholbach (talk) 13:52, 3 June 2017 (UTC)
It appears that the formula in Bourbaki simply has a typo (a ^3 missing), which I have now corrected and added another reference. Thanks for pointing this out! Jakob.scholbach (talk) 00:24, 4 June 2017 (UTC)
Ah, yes, now I see! Nice. Boris Tsirelson (talk) 06:30, 4 June 2017 (UTC)

"Topological fields": "The algebraic closure Qp carries a unique norm..." – probably, ${\displaystyle {\overline {Q}}_{p}}$ rather than ${\displaystyle {\overline {Q}}_{\overline {p}}}$ ? Really, I do not know how to implement it with the latex-free notation. But I try: Qp. Success? Boris Tsirelson (talk) 08:30, 4 June 2017 (UTC)

It could be debatable that Qp is actually a better notation, since we take the algebraic closure of Q_p, and don't apply some operation (depending on p) to \overline Q. However, I agree your suggestion is more standard, so I switched to it. Jakob.scholbach (talk) 18:07, 4 June 2017 (UTC)
Then maybe you prefer ${\displaystyle {\overline {Q_{p}}}}$ ? Boris Tsirelson (talk) 19:59, 4 June 2017 (UTC)
I don't care much. (I do care much more about the painful markup, if you are interested, weigh in at WT:WPM in a discussion about MathJax.) Jakob.scholbach (talk) 09:17, 5 June 2017 (UTC)

"Local fields": "In this relation, the elements p ∈ math and..." – probably, pQp and ... ? Boris Tsirelson (talk) 08:35, 4 June 2017 (UTC)

Of course. Jakob.scholbach (talk) 18:07, 4 June 2017 (UTC)

"Galois theory": "the roots of f can not be expressed in terms of addition, multiplication, and roots" – well, I do understand, but I feel some discomfort because of the two different meanings of the word "root" in one phrase. And by the way, the next phrase contains "the zeros of the following polynomials"... And in "Abel–Ruffini theorem" I see "there is no algebraic solution—that is, solution in radicals". Boris Tsirelson (talk) 11:17, 5 June 2017 (UTC)

"polynomial in E(a0, ..., an−1" – rather, in E(a0, ..., an−1). Boris Tsirelson (talk) 11:33, 5 June 2017 (UTC)

"Invariance of fields": "defined as the number of elements in F which are algebraically independent" – I'd say, "...as the maximal number of..." And again "Qp". Boris Tsirelson (talk) 11:42, 5 June 2017 (UTC)

"Model theory of fields": I feel some discomfort seeing text of the form "A if and only if B if and only if C" (even though formula "${\displaystyle A\Leftrightarrow B\Leftrightarrow C}$" is OK with me). I usually write "the following three conditions are equivalent". Boris Tsirelson (talk) 11:54, 5 June 2017 (UTC)

All done. I have chosen to avoid the word "equivalent" in stating that a sentence holds in C iff it holds in high characteristic, since the word "elementarily equivalent" might cause a minor source of confusion at this point. Jakob.scholbach (talk) 12:04, 5 June 2017 (UTC)

"and using Steinitz' theorem, this shows" – wow... I do not say otherwise, I just bother, since the linked "Steinitz' theorem" seems to deal with something else... really, so? Boris Tsirelson (talk) 12:02, 5 June 2017 (UTC)

"The absolute Galois group": "are not algebraically (or separably closed)" – does it mean, not algebraically (or separably) closed? Boris Tsirelson (talk) 12:06, 5 June 2017 (UTC)

"K-theory": ${\displaystyle x\otimes 1-x}$ looks for me worse than ${\displaystyle x\otimes (1-x)}$ (seen in our "Milnor K-theory" article). Boris Tsirelson (talk) 12:16, 5 June 2017 (UTC)

"Applications"

"Geometry: field of functions": "X can be reconstructed from k(X)" – up to... what? Boris Tsirelson (talk) 12:33, 5 June 2017 (UTC)

All done. Thanks again! Jakob.scholbach (talk) 12:50, 5 June 2017 (UTC)

(The end. Being not an algebraist I did not understand all of it, of course. But I do feel, vaguely, that it is exciting mathematics.) Boris Tsirelson (talk) 12:52, 5 June 2017 (UTC)

Thank you! Jakob.scholbach (talk) 13:11, 5 June 2017 (UTC)

## constructible numbers

Rather than just citing that arbitrary square roots can be constructed, it might be helpful or more transparent to readers to explicitly mention the Euclid's geometric mean theorem or the Pythagorean theorem (cathetus property). Maybe even give a (2nd) illustration, commons:Category:constructible number already has a bunch.--Kmhkmh (talk) 14:58, 1 June 2017 (UTC)

You mean commons:Category:Constructible number. Boris Tsirelson (talk) 15:09, 1 June 2017 (UTC)
Yes, thanks. For some reason the old link wasn't displayed red.--Kmhkmh (talk) 16:26, 1 June 2017 (UTC)
I have replaced the construction of products by square roots which illustrates better the specific properties of this field. (I think two illustrations would give too much weight for this one example in this article.) Jakob.scholbach (talk) 21:23, 1 June 2017 (UTC)
I can see why 2 illustrations might be too much, but I don't understand the particular choice of the current illustration (a general illustration of the chord theorem or geometric mean theorem) as there are other illustration available explicitly showing the root (1, 2, 3, 4, 5 and 6). Also i'd still mention both theorems in the text and avoid the complex number plane as it is not needed here. Strictly speaking you have 2 different fields, the constructible real numbers and the constructable complex numbers (or constructible points in the plane) and for the purpose of the article it might be better to stick to the former rather than the latter.--Kmhkmh (talk) 23:25, 1 June 2017 (UTC)
I think the file I chose and the ones you suggest all have (minor) advantages and disadvantages, I currently think the first two below are the best ones. The only drawback of the first is, as you say, that a h²=pq statement (or h=sqrt(pq)) is not mentioned in the illustration. However that can be easily added in the image caption. So even though I admit I had not checked all the files in the category carefully, I still think the one I chose first is best, but I am not at all adamant about it. If you want, feel free to reply right in the table below.
 the labelling of the line segments can be easily used in an image caption, most general picture among the ones listed below in that q ≠ 1 is "allowed" slightly more special (q=1); the image caption can not refer to the height other than by saying "it is sqrt(p)"; minor point: different fonts the used cathetus property is, as far as I can see, one mathematical step more?, also in the full generality it requires a distinction by two cases (as is done below) few labels, in particular the midpoint is "missing" even less labels, no indication of 90° angles similar to above comment; the (necessary) distinction of two cases is in my opinion distracting from the given goal of constructing a square root line segments not labelled; fonts are a bit ugly (in my opinion)
About real vs. complex constructible numbers: I am not sure I agree with you: in my mind it is somewhat more natural to construct a point in the plane as opposed to always projecting it back to the real axis. What are, in your opinion, advantages and disadvantages of introducing these two variants in this article? Jakob.scholbach (talk) 06:32, 2 June 2017 (UTC)
I rather disagree with your approach to the picture. Yes the first pictzre is the most general one with the most labels, but that is why you don't to use it. As rather than simply constructing a line segment of length ${\displaystyle {\sqrt {d}}}$ from a line segment of length ${\displaystyle d}$, it just solves a different more general problem in which a root appears. As far as the different fonts in some of the pictures are concerned, I'd say they make sense, as they are associated with different types of objects (points versus distances).
Yes, complex numbers a more natural for describing locations in the plane. But that again is imho an unnecessary more general problem as the original one. Instead of looking at constructible distances/lengths of line segments (the original Greek problem) we look at coordinate system and constructable points. The reader needs to have (slightly) more background knowledge for that, has to deal complex multiplication, and the commonly used inclusion chain integers - rationals - constructible numbers - algebraic numbers - reals doesn't really hold, but you need to use the complex counterparts.--Kmhkmh (talk) 08:26, 2 June 2017 (UTC)
root construction
root construction
Since you have created many of these files: could you tweak version 2 above so that it includes "q=1" and "h=sqrt(p)" (instead of just 1 and sqrt(p))? Such a tweaked version would be optimal for this article, I think. The point about the labels is this: it is easier to refer to the line segments in the text when there are the labels, otherwise we have to introduce notation for some line segment and the length of it, which is altogether unnecessary of there is a simple label in the diagram. I do want such labels.
Real / cx: The point is that in this article we neither prove (nor indicate much) why the field in question actually is a field. The purpose here, IMO, is to show that there are other, "less usual" examples of fields. Since the constructions really yield points in the plane, this is, IMO, more natural to understand for the unitiated reader. Otherwise we have to say "we only look at the lengths of the line segments we have constructed", which is also elementary and easy enough, but I think adds an unnecessary twist.
We can even remove the phrase "complex numbers" in this paragraph altogether. I don't think knowledge of complex numbers is necessary to understand "complex constructible numbers". It does require a bit more work to actually show these "complex construtible numbers" form a field, but since we don't do this job here in any case, presenting the example which can be formulated in the easiest (and, it turns out, most general) way is best, I think. Jakob.scholbach (talk) 10:46, 2 June 2017 (UTC)
OK, if we are to describe the field operations on the set of cx. constructible numbers, we essentially delve into the definition of these operations on complex numbers.
Why don't you tweak the section as you see fit, and we compare with what we have now? Jakob.scholbach (talk) 10:50, 2 June 2017 (UTC)
Options for the suggested graphic.--Kmhkmh (talk) 21:27, 2 June 2017 (UTC)
Thank you! Both are better than what we currently have. If it is not too much of a hassle, could you rotate the second one by 90°? I believe it is somewhat more standard (even though factually irrelevant) to have the height going vertical. Jakob.scholbach (talk) 13:18, 3 June 2017 (UTC)
commons:File:Root construction geometric mean5.svg --Kmhkmh (talk) 14:19, 3 June 2017 (UTC)

I have implemented the changes, using your picture. Thanks! Jakob.scholbach (talk) 00:41, 4 June 2017 (UTC)

## main article tag for (summarizing) sections/chapters

For many sections it might be appropriate to add an {Main article|<target article>}} template, rather than just linking a keyword in the normal text. In my experience that is pretty much standard for larger summary/overview articles and it provides for an easier navigation for readers.--Kmhkmh (talk) 15:03, 1 June 2017 (UTC)

I realize that the article is a bit inconsistent in this regard: in some cases we have the main article template, in other cases we just italicize the target of the would-be main article template. The latter option is, IMO, less obtrusive to the reader and permits a smoother reading. I don't have a strong opinion about this, but since the main-article-target is in many cases just mentioned in the very first sentence or paragraph of the section, I don't see much benefit in adding a main-article template. If you do, just go ahead and add it where you see fit, though. Jakob.scholbach (talk) 21:27, 1 June 2017 (UTC)

## Arithmetic modulo 12

This isn't actually wrong, but it may well confuse readers. In the section Field_(mathematics)#Finite_fields, there's a diagram about "modular arithmetic modulo 12". But unlike say modular arithmetic modulo 7, that doesn't give rise to a field. Maproom (talk) 12:05, 4 June 2017 (UTC)

Perhaps the relevant sentence This construction yields a field precisely if n is a prime number. , which is already contained, could get some emphasis to attract more attention. Purgy (talk) 15:29, 4 June 2017 (UTC)
I have added a clarifying remark in the image caption. Jakob.scholbach (talk) 18:08, 4 June 2017 (UTC)

## GA Review

Reviewer: Tsirel (talk · contribs) 13:02, 5 June 2017 (UTC)

Well, this is a math article. Not recreational mathematics. Not a textbook. Not a pearl of popular science. To a reasonable extent, it does contain elements of these three genres.

The article is clearly written and nicely organized. It is factually accurate; no original research; based on reliable sources. It covers broadly the topic without unnecessary digressions. Written from the neutral point of view. Stable, with no ongoing edit wars. With appropriate images. Without obvious copyright violations.

"Ideally, a reviewer will have ... sufficient expertise to verify that the article reflects the content of the sources; this ideal is not often attained" (quoted from WP:RGA#Assessing the article and providing a review). Indeed, I am a mathematician but not algebraist; my expertise is not sufficient for sections 7.2.1, 9.2, 9.3, 10.4, 11. Additional opinion of an algebraist is welcome. Nevertheless I am bold enough for claiming that this article is good! Boris Tsirelson (talk) 14:10, 5 June 2017 (UTC)

Thank you! Jakob.scholbach (talk) 14:58, 5 June 2017 (UTC)

## question on section "characteristic"

This is a Question on the short section "Characteristic":

Is n in F?

Remark: Reading this section makes it seem like n is in F. But this section doesn't state that, it just says n is a positive integer. --2A02:8108:1A00:3000:6444:6F79:5D49:E99F (talk) 03:41, 18 June 2017 (UTC)

n is (usually) not in F. n just has to be a positive integer. Note that it is (in my opinion) a (convenient) abuse of notation to use the expression ${\displaystyle n\cdot x}$ to denote both scaling by an integer (as in all Z-modules) and field multiplication.--Jasper Deng (talk) 03:58, 18 June 2017 (UTC)
Just to make life more complicated, for a positive integer n, one could write ${\displaystyle n\cdot 1=n}$ and it becomes apparent why this is a notational abuse. The n on the left is a positive integer, while the n on the right is a field element (being the sum of 1's). Moreover, depending on the field, the two n's may not even be the same integer, i.e., ${\displaystyle 7\cdot 1=2}$ in F5. --Bill Cherowitzo (talk) 05:07, 18 June 2017 (UTC)
There were several small attempts to make this abuse of notation more explicit, which were removed for the sake of stringency. Consider the re-inclusion? Purgy (talk) 06:03, 18 June 2017 (UTC)
I have included a remark that this operation is (formally speaking) distinct from the product of two elements of the field. Jakob.scholbach (talk) 12:35, 18 June 2017 (UTC)

## Question on section "A field with four elements"

The following discussion is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

If the picture in this section is accepted as true, the equation in this section is true. I can't see why one should accept this table/picture as true based on just the knowledge one gains from reading the above sections, i think thats not good. maybe one is required to read the links which are in the text above this section?

I can confirm that "O I A B" (in first row in + picture, in first column in + picture, in second row in * picture, in second column in * picture) is true. this confirmation is based on the above Fieldaxiom stating that the additive and multiplicative identity if paired in a binary operation with another operand does "give back" the other operand. All other results seem to somehow appear out of the blue. Am i overlooking something? Thanks in advance. --2A02:8108:1A00:3000:3C60:1D15:C104:D5A6 (talk) 23:58, 20 June 2017 (UTC)

The difficulty you are having stems from the fact that this field with four elements is being presented as an example–so the addition and multiplication tables are presented, but how they are constructed is not mentioned. All you can do in this circumstance is verify that the field axioms hold, you can not prove that they hold (except by exhaustively verifying all possibilities) or predict what the entries in the tables should be. If you want to do these things you need to know how the tables are constructed. These tables are constructed in the article finite field, but I should warn you that the construction is sophisticated and not a simple thing, which is probably why the construction details are not given in this article. --Bill Cherowitzo (talk) 02:48, 21 June 2017 (UTC)
Answering from a slightly other viewpoint, please, take the tables as defining the required field operations, and consider it your task to prove that they satisfy the required properties. From a (deeper) result on this kind of fields, this is the only way to set up appropriate tables, when disregarding simple renaming of the 4 elements.
In an effort to construct the tables you might want to consider the easy to see properties of the two unities (O, I), which, together with commutativity, determine already two rows and columns, and speculate just on the remaining 4 fields. Purgy (talk) 06:26, 21 June 2017 (UTC)
• It is straightforward to verify the field axioms for this field (whose operations are defined by the tables, as mentioned in the previous comment) by simply looking all the possible cases. If you want a derivation, you'll have to read up on quotient rings and polynomial rings, and then consider the quotient of the polynomial ring of the field with two elements by the ideal generated by a certain polynomial. That derivation is left as an exercise. If you'd like, I can help you try to understand it, but that belongs at Wikipedia:Reference desk/Mathematics, not here.--Jasper Deng (talk) 06:48, 21 June 2017 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.