Talk:Finite field

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Finite field has been listed as a level-5 vital article in Mathematics. If you can improve it, please do. This article has been rated as B-Class.

Too technical?[edit]

Would it be possible to change the first paragraph of this article to something that might be less formally correct, but more understandable to somebody who wants to learn about Galois Fields. The examples in Finite_field#Some_small_finite_fields were a step in the right direction. Chumpih (talk) 06:11, 9 March 2014 (UTC)

The article starts by saying that a finite field is a field which is finite. How can you make that less technical? Readers may have difficulties with the definition of "field", but that is not a problem for this article. Maproom (talk) 09:05, 9 March 2014 (UTC)

I believe that the point was not the first sentence, but the remainder of the first paragraph. I have rewritten the lead to be less technical. In paricular, I have removed the mentions of Brauer group (technical and unneeded), Frobenius endomorphism (out of scope, it is about algebras over finite fields), splitting field (not needed and confusing here). I have also removed the "recent results", as mentioning them here give them an undue weight. The same for the areas of applications, which are either subareas of already cited areas or are minor subjects sourced only by primary sources.

Question: Is the last paragraph of the lead really useful (the chain of algebraic structures that may be specialized to general field? D.Lazard (talk) 11:40, 9 March 2014 (UTC)

I consider the lead now greatly improved. As for its last paragraph - yes, I consider it useful, even though I have no idea what most of the things in the chain are. I can understand what it is saying, even though I have no idea at all what an "integrally closed domain" is. But I do wonder if the paragraph really belongs in this article. All that really matters is that a "finite field" is a field that is finite. The chain also appears in the field aricle, which is the proper place for it. Maproom (talk) 22:26, 9 March 2014 (UTC)

I think the benefit of the inclusion chain here is limited. There is the same chain under Field (mathematics) to which this links, and the name "finite field" is so suggestive that someone trying to understand the concept would find it with little delay. I'd think it would be cleaner not having it there, but I have no strong feeling on the matter. —Quondum 23:35, 9 March 2014 (UTC)

I came here from Reed–Solomon error correction trying to figure out what the referenced "finite field" is. This article is nearly incomprehensible to anyone who does not already know exactly what a finite field is. Oh, but its just a field which is finite, you say. Ok, so what a field? Ah, its just a non-zero commutative ring. Well now I'm getting somewhere! Just a few more clicks and I learn that a commutative ring is a kind of ring. Not sure what that is, but helpfully the article tells me its simply a kind of abelian group- and everyone knows what that is! So now I just have to read the full contents of the page on group theory, abelian groups, commutative rings, algebraic fields and about 10 more, all written for an audience of grad students, before I can arrive at the intuitive explanation of "a finite field is a mathematical object consisting of a finite set of elements, having defined the operations of multiplication, addition, subtraction and division between any two elements in the field in terms of a third element in the field." — Preceding unsigned comment added by 38.88.177.106 (talk) 18:30, 30 April 2014 (UTC)

You have done impressively well in your learning what the term means. I can't fault your final sentence. But I don't know what could be done about the problem. I guess it's rare for anyone without university entrance-level math to even come across finite fields. Maproom (talk) 19:34, 30 April 2014 (UTC)

A one-sentence encapsulation as is given here would be a useful addition to the lede, adding that division by zero is excluded. When a lede can sensibly give a brief understandable encapsulation in layman's terms, it should. —Quondum 19:48, 30 April 2014 (UTC)

I came to this article to try to understand the theory behind Reed-Solomon error correction. I'm an engineer, not a mathematician and though I have A-Level mathematics I didn't study the subject to undergraduate level, but nevertheless I feel that with a bit of effort I ought to be able to understand this article. I'm afraid I find it complete gibberish and I can't help but wonder if there's some form of elitism at play here. Could someone who understands the subject and who sees the value of teaching intelligent people who want to learn make a little effort to simplify it, please, in order to make it more accessible? This is not how a general encyclopaedia should be. I learned much more by reading the documentation for the implementation of RAID-6 in the Linux kernel - an application of Reed-Solomon and a practical use of Galois fields. 83.104.249.240 (talk) 16:50, 12 December 2015 (UTC)

I think the problem is really in the article field. I like the definition offered above: "a finite field is a mathematical object consisting of a finite set of elements, having defined the operations of multiplication, addition, subtraction and division between any two elements in the field in terms of a third element in the field" (except that it's not totally accurate - division by zero is undefined), but would prefer to list the operations in the order addition, subtraction, multiplication, division. Maproom (talk) 11:08, 13 December 2015 (UTC)

I agree that the lead of Field is much too technical. However I do not see any way to make the lead of Finite field less technical. Surely, a large part of the lead and the article are devoted to properties that are useless for studying Reed-Solomon codes. But they are to be there, as they are fundamental, and are widely used in code theory (for example Goppa codes and Elliptic curve cryptography, which is used in your connection to Wikipedia, if your computer is not out of date). As far as I know, the main properties of finite fields that are used for Reed-Solomon codes is that they are finite and that the vector spaces over them are also finite (if their dimension is finite). Therefore, the information that you are looking for is probably in Vector space and/or Polynomial ring. In particular the theory behind cyclic codes is strongly related with [[Polynomial ring#Quotient ring of K[X]|Polynomial ring § Quotient ring of K[X]]]. D.Lazard (talk) 16:31, 13 December 2015 (UTC)

Inclusion chain[edit]

The inclusion chain should probably be moved to a related structures section (somewhere toward the end of the article) or to a sidebar. This is how it's done in most other similar articles around here because it's not incredibly essential info and is less intrusive that way. Some1Redirects4You (talk) 17:04, 2 May 2015 (UTC)

I agree, and implemented this suggestion a month ago. Since it was recently reverted, I defend the edit here. The chain of inclusions is almost totally irrelevant to finite fields; no one studies finite fields as examples of commutative rings or UFDs. Putting the bulky related structures list in the lead over-emphasizes these not very important containments; the lead is already very long without them. If these containments are not important enough to be discussed in the body of the article, they should not be in the lead. These points are true regardless of how any other articles may be written. --JBL (talk) 16:06, 20 January 2016 (UTC)

I fully agree with Joel. From some points of view it might be more appropriate to consider them as FINITE division rings, putting more emphasis on the finiteness and downplaying the commutativity. The inclusion chain is just one of many ways of looking at finite fields and certainly does not belong in the lead. Bill Cherowitzo (talk) 19:04, 20 January 2016 (UTC)

Thanks for clarifying, I wasn't aware of this discussion here. (Wikipedia is so Web 1.0 disorganized.) Oftentimes my entire purpose of visiting one of these articles is just to review the chain, but I agree that this chain is just one of many views. It does seem to me that each of the articles should match in structure (for consistency of experience / improved reader-interface design). (I added it to the top because I was surprised that "this article is part of the chain template, but the template is not referenced in the article - based solely on my expectation from the other articles. I only happened to find it at the bottom a few minutes later from a related Google search while looking for something else.) --Yoda of Borg (✉) 11:04, 21 January 2016 (UTC)

It would be great if related articles used the same structure. And if they used the same notation. I don't see either happening any time soon. Maproom (talk) 11:07, 21 January 2016 (UTC)

I'd make it happen right now if I weren't lazy and so very tired. --Yoda of Borg (✉) 11:14, 21 January 2016 (UTC)

Two examples for character, in the lead?[edit]

I'm guessing this is part of the valiant effort to make the article more accessible. I disagree that two examples of that arithmetic are needed in the lead though. Some1Redirects4You (talk) 17:04, 2 May 2015 (UTC)

"Abstract algebra" or "algebra"[edit]

I have modified the first sentence of the article to change "abstract algebra" into "algebra", because the term "abstract algebra" was too restrictive. In my edit summary, I have given the example of cryptography, which can hardly be qualified as a part of abstract algebra, although the parts of cryptography that use finite fields belong to algebra.

My edit has been reverted by MvH with the edit summary In the US, "algebra" usually refers to high-school math, anything to do with rings/fields is called "abstract algebra" in the US even if it is very basic. Firstly this means implicitly that this article concerns only US readers, and not mathematicians, for which algebra includes also advanced questions. Secondly, this assertion implies also that "algebra" would be a synonymous of elementary algebra. If this would be true, why having separate articles? My opinion, which is certainly shared by most professional mathematicians, is that "abstract algebra" is the study of the algebraic structures for themselves, while "algebra" includes applications and interactions with other areas of mathematics (such as the relationship between commutative algebra and algebraic geometry).

I was ready to revert MvH revert, when I have realized that even "algebra" is too restrictive. In fact, the term "In ...", which is standard in the first sentence of mathematics articles, is not aimed to categorize the article, but to inform the reader if he could be interested in the article. As a reader may come here from cryptography, from number theory and other fields of mathematics, which are not commonly included in algebra, this beginning must be enlarged. Therefore, I'll replace "algebra" by "mathematics". D.Lazard (talk) 10:12, 7 May 2015 (UTC)

To D.Lazard: The naming system is like this in the US (yes, I agree it is a bit counterintuitive, I'm sure the terminology is different in France).

(1) "Solve ax^2+bx+c = 0": algebra

(2) Basic polynomial rings/groups/fields: abstract algebra (undergraduate)

(3) Modules over a PID: algebra (graduate)

(4) Topics you would consider abstract algebra: abstract algebra (graduate)

Most mathematicians would consider (1) = basic math, (2) = basic algebra, and (3) = algebra. But to see how the terminology is used in textbooks, look up "Thomas Hungerford" on amazon, and you see there that the basic topics (2) are in a book called "Abstract Algebra" whereas more advanced topics (3) are in a book called "Algebra". So his "Abstract Algebra" book is much more elementary than his "Algebra" book. To a European, I'm pretty sure this comes as a surprise, but here, the way you can see on the cover which of the two books is the elementary one is by looking for "undergraduate text" or "graduate text". The phrase "abstract" is not used to indicate the level, it's used to indicate the topics (i.e. it's used to indicate topics that don't resemble (1)). The terminology "abstract algebra" for basic stuff like finite fields (mathematicians wouldn't consider this abstract) is wikipedia-correct (i.e. supported by references). Moreover, I think other wikipedia pages also use the phrase "abstract algebra" in this way (I expect that other languages don't use the term this way). MvH (talk) 12:28, 7 May 2015 (UTC)

I disagree with the argument of broadening the discipline of the study of finite fields to include cryptography. Cryptography is an application of abstract algebra, just as medicine is an application of NMR through MRI. I think most cryptographers would be happy to admit that (some of them) use finite groups, finite fields, lattices and other algebraic structures because they have found that they exhibit useful complexity properties, but to conclude that cryptographers consider groups or fields as an area of study in their own right would be pushing it too far. It would be like saying that vector spaces are a field of engineering. —Quondum 13:21, 7 May 2015 (UTC)

I never said that "finite fields" include (or should include) cryptography or number theory. I said that the sentence "In <some area>, a finite field is ..." must not mean that <some area> is the area which studies finite fields in their own right, but must denote the area in which the definition that follows is unambiguous and widely used. I guess that nobody will disagree that finite field means the same thing for a cryptographer, a number theorist, an abstract algebraist and more generally for any mathematician. Therefore <some area> must, here, be "mathematics". However, I agree with the fact that category:finite fields is a subcategory of category:abstract algebra, which is itself a category of category:algebra, but this is not the subject here.

Apparently, for MvH, there is a discrepancy between terminology of US mathematics teachers and that of the researchers in mathematics (in USA as well as outside USA) (note that Serge Lang never uses "abstract algebra" in his famous book Algebra). IMO, this US teaching terminology is well explained by history, but it is somehow old fashioned, at graduate level, as well as at undergraduate level in many countries. These historical reasons are the following: until the end of 19th century, "algebra" was the manipulation of symbols as they were numbers, and also the theory of equations (example (1) of MvH). A the turn of 20th century, the algebraic structures have been introduced, and their studies have been called "abstract algebra" or "modern algebra". Later on, these algebraic structures became standard in every part of mathematics, and the distinction between classical algebra and modern/abstract algebra has been forsaken. A witness of this is the change of the name of van der Waerden's book from "Modern algebra" to "Algebra". As "abstract algebra" is still widely used by US teachers, I agree to keep the article, but, IMO, the term "abstract algebra" must be used only in the rare cases where the distinction between "algebra" and "abstract algebra" is really relevant (probably only in mathematics education). D.Lazard (talk) 14:44, 7 May 2015 (UTC)

I agree with your viewpoint, but if there is a discrepancy between (a) best terminology, and (b) what is common in references, then wikipedia will go with (b) instead of (a). The book you mentioned is not a counter-example to what I wrote. On the contrary, it confirms what I wrote, the phrase "Graduate Texts" is written in large font on the cover of Lang's book, so it fits perfectly under (3) above. MvH (talk) 15:44, 7 May 2015 (UTC)

I disagree that the <some field> must cover the broadest field in which the the concept is widely used. In contrast, it should be the narrowest that contains the field of study and that is likely to be known to (or easily determined by) the typical reader of the article. For example, an engineer would like to know at first glance which subfield of mathematics it is studied in. Thus, if we want to keep the "In mathematics, ...", we should do as in Ring (mathematics), and include "more specifically ...". Otherwise, just use the more specific category. I'm happy with algebra, as I would be with abstract algebra.

I tend to disagree with the position that the term "algebra" being interpreted in a colloquial way in many parts of the world should steer its use here. Words have multiple, often related, meanings, and this is normal in WP. The reader will quickly discern that algebra is the name of a formal subfield of mathematics not the same as the name of the class at grade school, the link provides the needed disambiguation, and we should not avoid the formal use of the word in WP. If we avoid its use here, we are saying that the article Algebra should never be linked to, and we should always link to Abstract algebra. WP should use the term in a consistent way in any context. I don't tend to agree with MvH's assessment, but this is just my perception. To me, algebra should refer here to a specific field of mathematics, not simply be "how the word gets used by sources". Distinguishing between the use of the term at different levels of education seems to me to be inappropriate. —Quondum 16:02, 7 May 2015 (UTC)

Quondum, we could use the word "algebra" and use "abstract algebra" in the link. In algebra .... MvH (talk) 19:55, 7 May 2015 (UTC)

I do not have any strong feeling about which, partly because I still don't have a strong sense of the distinction yet. To me it feels like fields fit into abstract algebra (the axiomatic approach to algebraic objects, as defined in Algebra), though the piping suggestion seems a little easter-eggish. —Quondum 20:39, 7 May 2015 (UTC)

Personally, I would be fine with the three starts "In mathematics", "In mathematics and more specifically in algebra" and "In algebra". However, as everybody has an idea (possibly confuse) of what is "algebra", I prefer "algebra" alone; the reader, who has a unclear idea of what it is, may, as usual, follows the link. On this point, I agree with Quondum. On the other hand, I oppose to "abstract algebra", which may be confusing for readers interested in applications of finite fields. Even after having followed the link, they could think: "Oh I am not interested in abstract structure in their own right, thus this article is not for me." D.Lazard (talk) 16:26, 7 May 2015 (UTC)

Confusing Notation[edit]

From the text:

"Let F be a finite field. For any element x in F and any integer n, let us denote by n⋅x the sum of n copies of x. The least positive n such that n⋅1 = 0 must exist and is prime; it is called the characteristic of the field.

If the characteristic of F is p, the operation {\displaystyle (k,x)\mapsto k\cdot x} (k,x)\mapsto k\cdot x makes F a GF(p)-vector space. It follows that the number of elements of F is pn."

The last sentence makes no sense! What is k? what is x? what is n? — Preceding unsigned comment added by 122.167.139.98 (talk) 19:24, 24 September 2016 (UTC)

Yes, you're right, more terms could have been defined. I have attempted to clarify. (The current version still does not quite spell things out as much as a textbook might.) --JBL (talk) 21:49, 24 September 2016 (UTC)

Joel beat me to it, but I had written this explanation already so I'll just copy it here:

A field F of characteristic p contains a subfield isomorphic to GF(p) (= Z_p). We can view F as a vector space where the elements of F are the vectors and the elements of that subfield are the scalars. The situation is a little unusual in that the scalars here, being elements of F, are also vectors, so when we describe scalar multiplication we need to be a little formal to avoid getting the roles of these elements mixed up. Thus, scalar multiplication is described by associating an element of F to the ordered pair (k,x) where k is a scalar (in the subfield) and x is a vector (any element of F). The associated element is

k\cdot x

that is defined in the previous paragraph since k can be thought of as an integer. Now that we know that F can be viewed as a vector space, since it is finite it must have a finite dimension (size of a basis) and this dimension will be denoted by n. Every element of F (the vector space) can be written uniquely as a linear combination of the n basis vectors (for any fixed basis). As there are p choices of coefficients for each of these basis vectors the total number of linear combinations is pⁿ.

Now I'll just sneak over to the article to see what Joel wrote.--Bill Cherowitzo (talk) 22:23, 24 September 2016 (UTC)

P.S. What I wrote above is more like a textbook presentation and not really suitable for an encyclopedia entry. I gave it so that the issues that needed to be amplified in order to make the sentence more understandable could be identified. --Bill Cherowitzo (talk) 04:32, 25 September 2016 (UTC)

"repeated addition" or "repeated multiplication"[edit]

In the article, it says

   If the characteristic of  $F$  is  $p$ , one can define multiplication of an     element  $k$  of  $GF(p)$  by an element      $x$  of  $F$   $(k,x)\mapsto k\cdot x$  by choosing an integer representative      for  $k$  and using repeated addition.  This multiplication makes  $F$  into      a  $GF(p)$ -vector space. It follows that the number of elements of  $F$      is  $p n$  for some integer  $n$ .

I feel that it should be "repeated multiplication", rather than "repeated addition". Am I wrong? Jackzhp (talk) 11:17, 16 January 2018 (UTC)

Yes

3\cdot x

means

x + x + x

, a repeated addition. D.Lazard (talk) 12:16, 16 January 2018 (UTC)

Rename the Article[edit]

A finite field is a field with finite number of its domain set. A finite field does not have to be Galois field. A Galois field GF(p^k) can be deemed as one with its domain set to be vectors of integer, and it is the extension of a simpler finite field F(p). I think that it is a bad idea to put much of Galois field stuff inside the article titled with "Finite Field" Jackzhp (talk) 10:26, 10 October 2018 (UTC)

I guess that you are confusing Galois fields and Galois extension. In all available sources, "Galois field" and "finite field" are synonymous. I have added a hatnote to the article. D.Lazard (talk) 10:42, 10 October 2018 (UTC)

@D.Lazard: I am not at all convinced that the terminological question you address is relevant to this user's confusion. @Jackzhp: every finite field is of order p^k for some prime p and positive integer k, and all finite fields of the same order are isomorphic; thus every finite field is (isomorphic to) a Galois field. --JBL (talk) 11:52, 10 October 2018 (UTC)

Every finite field is (isomorphic to) a Galois field: such a theorem makes sense only if "Galois field" has been defined. I do not know any source giving a definition of a Galois field that is not "a field with a finite number of elements". Thus, the above quotation makes sense only with another definition of a Galois field. What is this definition? Do you have any reliable source for it?

On the other hand, it is true that every finite field is a Galois extension of a prime field, and that a Galois extension of a prime field is a finite field if and only if the characteristic is not zero.

Also, one should note that the phrase "Galois field extension", which I have used in the disambiguating hatnote, is intrinsically ambiguous, as it may be understood as a "field extension that is a Galois extension" or as a "field extension of a Galois field". D.Lazard (talk) 12:43, 10 October 2018 (UTC)

If you read Jackzhp's comment again, you will see that Jackzhp has some concrete model of Galois field in mind ("A Galois field GF(p^k) can be deemed as one with its domain set to be vectors of integer, and it is the extension of a simpler finite field F(p).") and so my comment directed to him is in reference to whatever this concrete model is. --JBL (talk) 13:47, 10 October 2018 (UTC)

Jackzhp wrote "A finite field does not have to be Galois field." Can anyone specify an example, a finite field that is not (isomorphic to) a Galois field? Maproom (talk) 13:55, 10 October 2018 (UTC)

To Maproom: Before giving examples, one requires a definition of a Galois field. As the only a available definition of a Galois field is to be a finite field, it is impossible to give such an example.

To Joel B. Lewis: I cannot understand the first part of Jackzhp's sentence, as there is no field structure on vectors of integers. My guess is that they have in mind that GF(p^k) is a vector space over the field of integers modulo p. This is true for every finite field, and thus does not help. The second part of the sentence ("is the extension of a simpler finite field F(p)" is also true for every finite field. D.Lazard (talk) 15:34, 10 October 2018 (UTC)

To D.Lazard:You said "'Galois field' and 'finite field' are synonymous",that is what I am against. I believe that's really wrong. And you mentioned Galois extension. Seems to me that you do not really understand the term, maybe I am wrong, but I believe that if you do then you would agree that what you called Galois field is indeed a kind of Galois extension. k=1 is the base field, then k>1 is the extension of the k=1 field. Maybe "vector" is not the right word, "series" should be used instead? it was talking about the dimension, a combination of several numbers as 1 element. Jackzhp (talk) 15:43, 23 October 2018 (UTC)

To Joel B. Lewis: Let's not to get into isomorphism when we are talking about the difference between Galois field and finite field. Jackzhp (talk) 15:43, 23 October 2018 (UTC)

Galois Field is not infinite, and it is a Field (mathematics), so it is a finite field. But this article should not get into Galois Field(F(p^k) with k>1) directly, as it is the extension of the simple Finite Field(F(p) with k=1). Someone mentioned example, I think we can make one easily: F(5) with set {0,1,2,3,4} and with addition and multiplication modulo 5. 2 is a quadratic nonresidue modulo 5, so an extension can be built on this, take u^2=2, u is an element in the set of F(5^2), the set is {0+0u,1+0u,2+0u,3+0u,4+0u,0+u,0+2u,0+3u,0+4u,...} total 25 elements. F(5^2) is a Galois Field which is an extension(exactly a kind of Galois extension) of F(5) by taking a quadratic nonresidue as basis. Does this example suffice? From F(5^2), we take some cubic nonresidue, we can do cubic extension, and get F(5^6), another Galois Field. Jackzhp (talk) 15:43, 23 October 2018 (UTC)

I just found a sentence from Field (mathematics) page:"The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions." I guess it might be written by someone specialized in math. I hope we all agree Galois Fields are fields extended from simpler field. Jackzhp (talk) 15:57, 23 October 2018 (UTC)

Wikipedia must not include original thoughts, and it is not the place for changing the commonly accepted terminology. The terminology and definitions appearing in this article appear in all textbooks on the subject. The fact that depending on the authors some different terminology may be used explains that we have different terminologies for the same thing. For example some authors talk of Galois fields, some others use finite field for the same concept. As your definition cannot be found in the literature, we cannot follow your ideas, even if the standard terminology may be confusing. D.Lazard (talk) 17:23, 23 October 2018 (UTC)

I fully support D.Lazard ! --Nomen4Omen (talk) 18:29, 23 October 2018 (UTC)

Not only is D.Lazard absolutely correct, but your so-called construction technique does not work. Ask yourself how you would construct the sequence of Galois fields of order $5 p$ for $p$ a prime. Your procedure is guaranteed not to produce anything new after, at most, the first four of these. Galois field and finite field are synonyms and have been since the terms were introduced in 1893 by E.H. Moore.--Bill Cherowitzo (talk) 20:10, 23 October 2018 (UTC)

To Wcherowi: my above example, 2 is a prime, 3 is also prime. why other prime p does not work? Any p you specify, I will get the basis for the extension field. Jackzhp (talk) 02:08, 24 October 2018 (UTC)

Sorry, I misread your example and have struck my comment. However, you have not made any case for a distinction to be drawn.--Bill Cherowitzo (talk) 05:55, 24 October 2018 (UTC)

To D.Lazard: take "Galois Field" and "Finite Field" as synonym is just a mistake, it is not correct. Though you are free to do so, and continue to do so, and continue to make freshman confused. Jackzhp (talk) 02:16, 24 October 2018 (UTC)

Jackzhp: You claimed above "A finite field does not have to be Galois field." I challenged you to provide an example of a finite field that is not (isomorphic to) a Galois field. You provided an example of a 25-element field which, in your own words, "is a Galois Field". It seems to me that we all agree that all finite fields are Galois fields, and all Galois fields are finite fields. I do not understand why you are trying to draw a distinction between the two terms. Maproom (talk) 07:45, 24 October 2018 (UTC)

To Maproom: For my example,I guess it is clear that my focus is that F(5) is a finite field, but not a Galois Field, while we say its extensions such as F(5^2), F(5^3), etc are the Galois Fields. I have said let's not to talk about isomorphism. Especially, "isomorphism" is called "isomorphism", is not called "synonym". If two structures are isomorphic, they are just "isomorphic", we still give them two different names. I think it is very clear when k=1, I prefer to call it （simple） finite field, while k>1, I call them Galois extension, or Galois Fields （they are still finite）. A basic finite field has only 2 properties: it is a Field, and it is with finite number. A finite field is a simple structure, there is no point to mess it up with many other more properties. If any structure with more properties, we better given them other names. If you want a finite field and people usually does not call it Galois Field, I can give another example: the field consists of rational polynomials while the coefficients are taken to be in a simple finite field. It is a field, and it can be finite（?）. See Rational function and Fraction field for more information. With your logic, since all finite fields with same order are isomorphic, do you call such a field a Galois Field? Or please prove rational function is not a field, or not it is not finite. If you count it as Galois field, then you will have to include Rational function & fraction field into this article. Jackzhp (talk) 03:19, 26 October 2018 (UTC)

I have to admit that the above example is not concrete enough. My experience is only with rational function as a finite group, I haven't think it through whether I can construct a finite rational function field. If you can prove that all rational function fields(fraction fields) can not be finite, it will be great, then I will know all rational function fields(fraction fields) are infinite. Jackzhp (talk) 03:43, 26 October 2018 (UTC)

For any field F, the ring F[x] of polynomials over that field is infinite: it contains as a (vector space) basis {1, x, x^2, ...}. The field of rational functions over the field is larger. If we take a quotient that is a field and is finite (like (Z/5Z)[ x ] / (x^2 - 3) ) then indeed it is both a finite field and a Galois field, because those two words mean the same thing. (Prime fields are field extensions of prime fields, too: they are extensions of degree 1. Just as R^1 is a vector space over R for every field R.) --JBL (talk) 10:17, 26 October 2018 (UTC)

To Jackzhp: You say I haven't think it through whether I can construct .... It is your own right to try inventing mathematics again. But trying this in Wikipedia is WP:disruptive editing, as Wikipedia is an encyclopedia, and, as such, it is aimed to present existing knowledge, not your WP:OR. Please stop your disruptive editing. I recall that disruptive editing may lead to be blocked for editing. D.Lazard (talk) 11:10, 26 October 2018 (UTC)

I don't get the hostility on display here; Jackzhp is not editing disrupively, they are discussing on the talk page ideas about the article, in a responsive way. If not all their ideas are correct, that does not mean they should be treated so unpleasantly. —JBL (talk) 11:15, 26 October 2018 (UTC)

I apologize for having been unclear. Opening this thread was absolutely not disruptive editing. But, here, there is a clear WP:consensus against the proposed renaming and the ideas that support it. Several experimented editors (including me), which are also professional mathematicians, have spent time for explaining why these ideas are wrong. Despite this, this editor does not consider the given answers and the resulting consensus, and repeats again and again the same rejected ideas. This is definitely disruptive editing, and I know several editors who have been topic banned or blocked for a very similar behavior. D.Lazard (talk) 14:02, 26 October 2018 (UTC)

Proposal for the 1st paragraph[edit]

A finite field is a field with finite number of elements for its set. A finite field can be a number field where its elements are numbers （might be multidimensional）, or a function field where its elements are functions such as rational polynomials. A simple single dimension number finite field can be extended to multidimensional ones which are called Galois extension and Galois Fields. Jackzhp (talk) 06:29, 26 October 2018 (UTC)

The first sentence is not better than that in the article. The remainder is WP:OR, and is completely wrong if one accepts, as all mathematicians do, the definitions given in Number field, Function field and Galois extension. It uses also undefined concepts, such that "rational polynomial", "multidimensional number" and "single dimension number". No further discussion is needed. D.Lazard (talk) 06:55, 26 October 2018 (UTC)

D.Lazard, you also do not have a perfect grasp of mathematical English, but no one gives this as a reason to refuse to discuss proposed edits with you.

Jackzhp, I think there is no chance that the article will be edited along the lines you suggest. One of the big triumphs of modern mathematics is the realization that particular labeling schemes are not important, what one needs to understand is the underlying structure (isomorphism type) and then all follows from that. You are proposing to turn this on its head by emphasizing the particular nature of the representation, and there is no possibility that you will find consensus for that. —JBL (talk) 11:33, 26 October 2018 (UTC)

To D.Lazard:: I did not touch the article. I hope we all agree that I have the right to express what I think. And whatever I wrote, I guess you are not required to respond or comment. Anyway, I really appreciate people like you who put continuous effort on editing wiki. At present, you are the person with power. What I can do is to assure you that I won't touch the article, so please don't get upset. Though I want to tell you that making Galois Fields to take the position of finite field really made me confused. Eventually I got it over: Galois Fields are finite fields, but they are not the simplest finite field. Jackzhp (talk) 15:21, 26 October 2018 (UTC)

To Joel B. Lewis::I totally agree with you about the structure. For long time, I have realized that mathematics is about structure. Hence, I guess when a concept is introduced and illustrated, the simplest one should be used, and then we introduce more complex ones and the morphism related stuff. Jackzhp (talk) 15:21, 26 October 2018 (UTC)

To Jackzhp: You say: «A finite field can be ... a function field where its elements are functions such as rational polynomials.» But so sorry: A field which contains polynomials can never be finite. Because there is some indeterminate which occurs in all finite powers. But if it occurs in all finite powers this means that it occurs infinitely often. --Nomen4Omen (talk) 16:51, 26 October 2018 (UTC)

To Jackzhp: I acknowledge that you have not edited the article. But Wikipedia rules apply also to talk pages. I recommend that you read (or read again) MOS:TALK, and in particular MOS:TALK#Maintain Wikipedia policy, which says ... it is usually a misuse of a talk page to continue to argue any point that has not met policy requirements.. Please, read also WP:NOTFORUM. D.Lazard (talk) 17:14, 26 October 2018 (UTC)

D.Lazard, your analysis of this interaction has been flawed from the very beginning (when you failed to understand the source of confusion in the initial message), and your relentless hostility towards this obviously good-faith contributor is completely inappropriate. If you do not think the discussion is productive, please take no part in it, and everything will be fine.

~~Nomen4Omen, please strike the completely unwarranted personal attack in your message; it serves no purpose.~~

Has the world gone mad that anyone thinks these kinds of behaviors are appropriate?! --JBL (talk) 17:37, 26 October 2018 (UTC)

List of small fields?[edit]

For groups, Wikipedia has the article List of small groups. Would a similar article for fields make sense? — Preceding unsigned comment added by 147.251.201.141 (talk) 21:51, 27 November 2018 (UTC)

No, the list corresponds to the list of primes and prime powers. Any q = pⁿ, where p is a prime number and n a positive integer, corresponds to a finite field F_q. There are many more axioms defining a field than axioms defining a group, so the variety of structures that satisfy the axioms is so much smaller for fields. — Rgdboer (talk) 02:46, 28 November 2018 (UTC)

Characteristic paragraph is a mess[edit]

The paragraph introducing characteristic is a total mess:

In a field of order $p k$ , adding $p$ copies of any element always results in zero; that is, the characteristic of the field is $p$ . Characteristic is defined in this way: If $F$ is a finite field, then for any element $x$ in $F$ and any integer $n$ , denote by $n \cdot x$ the sum of $n$ copies of $x$ . The least positive $n$ such that $n \cdot 1 = 0$ exists and is a prime number is called the characteristic of the field. The operation $n \cdot x$ can be defined as multiplication; If the characteristic of $F$ is $p$ , one can define multiplication of an element $k$ of $GF(p)$ by an element $x$ of $F$ $(k,x)\mapsto k\cdot x$ by choosing an integer representative for $k$ and using repeated addition. This multiplication makes $F$ into a $GF(p)$ -vector space. It follows that the number of elements of $F$ is $p n$ for some integer $n$ .

The order in which information is introduced is incoherent, and a proof of the prime power property dubiously belongs in a section called "Definitions, first examples, and basic properties". I do not have time to fix this right now, so I am leaving this note in case someone else wants to take a look before I do (which might not be for days or weeks). --JBL (talk) 16:52, 5 April 2019 (UTC)

This mess has been introduced by an edit dated on 21 February 2019‎. I have reverted it. D.Lazard (talk) 22:11, 5 April 2019 (UTC)

I also support removal of the sentence saying a field is defined with four operations. We don't distinguish between multiplication and division, or addition and subtraction; in each case the latter two are defined using inverse elements.--Jasper Deng (talk) 22:15, 5 April 2019 (UTC)

I have moved this sentence (with four removed) to its right place, ar the beginning of the section. There were also redundancies, which could be confusing for some readers. I have removed or edited them. IMHO, it is important that the main properties remain easily found at the beginning of the section. This is the reason for not removing all redundancies. D.Lazard (talk) 10:25, 6 April 2019 (UTC)

Proposal to add a section for isomorphisms aka composite fields aka sub-field mapping[edit]

I feel there should be a section about composite fields. A common example is mapping of GF(2^8) to GF(((2^2)^2)^2) for inversion (1/x) of a GF(2^8) number for AES encryption hardware where there may be 10 or more encoders/decoders on a single card, and the composite field greatly reduces gate count. Prior to carryless multiply instructions, composite fields allowed for faster computations for GF(2^32) and larger fields. However, with the carryless multiply pclmulqdq instruction on a X86 (since 2008), a GF(2^32) or GF(2^64) multiply can be done with 3 pclmulqdq and 1 or 2 xors. Rcgldr (talk) 22:58, 23 May 2020 (UTC)

There is also some little known information that should be archived somewhere, such as if a primitive field is going to be mapped to a primitive composite field, both using the standard primitive element x+0, then the larger field polynomial with coefficients changed to be the smaller fields coefficient size will have factors that correspond to the mappable fields (using alternate primitives allows for more flexible mapping). For example, for GF(2^32) = x^32 + ^x^22 + x^2 + x + 1 mapped to GF((2^16)^2) x^2 + ax + b with GF(2^16) coefficients based on x^16 + x^12 + x^3 + x + 1, if GF(2^32) polynomial reinterpreted as GF((2^16)^32) = x^32 + ^x^22 + x^2 + x + 1, then the 16 factors of GF((2^16)^32) in the form x^2 + ax + b, will be the mappable polynomials (out of a possible 4 billion). Rcgldr (talk) 22:58, 23 May 2020 (UTC)

How the matrices used to map between a primary field and a composite field are generated is difficult to find. Rcgldr (talk) 23:01, 23 May 2020 (UTC)

Can you point to any appropriate sources that discuss this? (At the level of mathematics, the objects GF(2^8) and GF(((2^2)^2)^2) are not different, though obviously the second notation emphasizes a particular idea of building it by three quadratic extensions. It seems plausible that there might be advantages of one representation versus another in CS, but we need reliable sources for that.) —JBL (talk)11:21, 24 May 2020 (UTC)

As mentioned above, there's been a lot of effort put into reducing gate count for AES S boxes, via GF(2^8) to GF(((2^2)^2)^2). Example references: GF(((2^2)^2)^2)(AES) S-box , A Very Compact S Box pdf. Other examples are about reducing large fields GF(2^n) for n >= 32: Efficient Software Implementations of Large Finite Fields pdf, which describes mapping GF(2^32) to GF((2^16^2), defines GF((2^16^2) and GF(2^16), but doesn't define GF(2^32) (it's probably jerasure standard x^32 + x^22 + x^2 + x + 1), and doesn't define the mapping matrix or it's inverse. I had to do an optimized brute force search to determine the missing data, to answer a question at a forum GF(2^32) to GF((2^16)^2) mapping. Rcgldr (talk) 12:50, 24 May 2020 (UTC)

As for generating mapping matrices, let k = m · n, and GF(2^k) is to be mapped to GF((2^m)^n). Let e = the number of elements to map at a time, formatted as a matrix, elements[k][e]. Define the GF(2) mapping matrix as map[k][k]. Mapping e elements from GF(2^k) to GF((2^m)^n) is done via map[k][k] x elements[k][e]. The inverse of map is used to map back. The column indexes (right to left) of map[][] correspond to bit indexes of elements, map[][{...,4,3,2,1,0}] correspond to element bits {...,4,3,2,1,0}. Let α(x) = a primitive element of GF(2^k), and β(x) = a primitive element of GF((2^m)^n). The column values of map[][{...,4,3,2,1,0}] are pow(β(x),log_α(x)({...,16,8,4,2,1})). I'm looking for a reference for this, but generally the articles I find assume the reader is aware of this and just show an image of the k by k bit matrix. Rcgldr (talk) 12:50, 24 May 2020 (UTC)

For a given GF(2^k), GF((2^m)^n) has to be determined to follow two basic rules. Let a and b be elements of GF(2^k), then while operating in GF((2^m)^n) map(a + b) = map(a) + map(b) and map(a · b) = map(a) · map(b). In the case where α(x) = x + 0 and β(x) = x + 0, GF((2^m)^n) can be any of the m factors of GF(2^k) reinterpreted to have coefficients of GF(2^m), essentially GF((2^m)^k). For better optimization, α(x) can be any primitive element of GF(2^k), and an optimized brute force for the "best" combination of α(x) and GF((2^m)^k) is done. This is described in the paper I linked to before: A Very Compact S Box pdf. Rcgldr (talk) 12:50, 24 May 2020 (UTC)

As far as I understand your post, you are concerned by computation is finite fields. For this, a representation of the field is needed, either as polynomials over GF(2) modulo an irreducible polynomial, or as a tower such representations. Your problems seems 1/ to determine the representation that leads to the most efficient computation 2/ (possibly) getting efficient algorithms for changing of representation. Both problems could be considered in the article (for the representation by a simple extension, the best choice of the defining polynomial is already sketched (with a reliable source) in section "non-prime field". For describing the known solutions of the other problems, we need sources. Those that you provide are not convenient for Wikipedia, as they are original research and consist mainly in proposed solutions, and they do not provide any synthesis of the knowledge in the subject. We need absolutely such a synthesis because, otherwise the presentation would be biased by giving emphasis to results for which we do not know whether they are generally accepted solutions. This is for this reason that original research and original syntheses are forbidden in Wikipedia (see WP:OR). As this subject seems to evolve rather quickly, this seems to early for presenting it in WP. D.Lazard (talk) 17:55, 24 May 2020 (UTC)

"Too early?" - there have been hardware implementations of composite (sub-field) mapping since the 1990's. I have a 1995 report from E J Weldon Jr for implementing hardware inversion (1/x) for GF(2^8) x^8+x^7+x^2+x+1 (hex 187, a primitive polynomial) using composite field (AKA sub-field mapping AKA "residue over a polynomial of degree 2 over GF(2^4)"). Usage of composite fields was fairly well known by the people involved in Reed Solomon codes or AES encryption by that time and probably dates back another 5 to 10 years prior to that report. In his 1995 report, he describes a mapping to GF((2^4)^2) x^2 + 4x + 2. Other implementations further reduced gate count by mapping to GF(((2^2)^2)2) and using a quadratic of the form x^2 + x + c. Some of the references in A Very Compact S Box pdf should qualify as reliable sources. Rcgldr (talk) 09:20, 25 May 2020 (UTC)

Again, the references that you provide do not allow writing the section that you are asking for, without doing WP:original synthesis. By "too early", I meant that we must wait a published synthesis on the subject, which is apparently still lacking. D.Lazard (talk) 09:41, 25 May 2020 (UTC)

I still am not getting what you mean by published synthesis. As for the algorithms and math involved, the articles are not lacking. The articles are out there, and I'm currently looking for such articles. I've found a few which I've posted below. Rcgldr (talk) 03:55, 26 May 2020 (UTC)

Here is a link to a portion of that report by professor E J Weldon Jr Finite Field Isomorphisms - Inversion pdf. I found more complete examples and explanations of using isomorphism for calculating a multiplicative inverse in GF(2^8). Starting at section E in aes sbox pdf and section 4.2 in Compact Rijndael Hardware Architecture pdf. Perhaps this should be also be added to Finite field arithmetic#Multiplicative inverse Rcgldr (talk) 16:56, 25 May 2020 (UTC)

Composite or sub-mapped fields are isomorphic in addition and multiplication: map(a + b) = map(a) + map(b) and map(a · b) = map(a) · map(b). This is explained starting at section 7.8.2, page 95 of Standford - Introduction to finite fields pdf. For binary field mapping of the form GF(2^k) to GF((2^m)^n), generally GF(2^k) is fixed, the parameters for GF((2^m)^n) are pre-selected, and a search is done for a primitive element of GF(2^k), which is used to generate the mapping matrix, so that the resulting field GF((2^m)^n) is isomorphic to GF(2^k) in addition and multiplication. In the case of minimizing gate count for a given GF(2^8) such as AES x^8 + x^4 + x^3 + x + 1, nested mapping may be used, such as GF(2^8) to GF(((2^2)^2)^2), and a search for a primitive element of GF(2^k) as well as some of the constants for GF(((2^2)^2)^2) is done to result in a minimal gate count. This is the focus of the "very compact s box" articles. Rcgldr (talk) 03:55, 26 May 2020 (UTC)

I now think this would be more appropriate in Finite field arithmetic, since the purpose is to optimize mathematical operations on finite fields. I'm dropping the proposal for here. Rcgldr (talk) 00:45, 27 May 2020 (UTC)

Notation[edit]

I'd like to propose that the article favor the notation $\mathbf {F} _{q}$ over the older and more cumbersome $\operatorname {GF} (q)$ . Both are used in the literature, so of course both should be mentioned in the article, but the current editions of most books by experts (Artin, Bourbaki, Lang, Serre, etc.) use $\mathbf {F} _{q}$ , so it would be good if Wikipedia followed suit by using $\mathbf {F} _{q}$ in most places, while mentioning $\operatorname {GF} (q)$ as an alternative notation. Ebony Jackson (talk) 19:52, 13 December 2020 (UTC)

IMO, notation

\mathbf {F} _{q}

should be discarded in favour of

\mathbb {F} _{q}

and

\operatorname {GF} (q).

The reason is that contrary to the two other notations,

\mathbf {F} _{q}

is ambiguous as it may denote an indexed variable (boldface is often used for variables representing vectors or matrices). When reading an article that does not define its notations, if one encounters

\mathbb {F} _{q}

or

\operatorname {GF} (q)

, one knows immediately that this denotes a finite field. On the contrary, with

\mathbf {F} _{q}

one needs to look at the context. So,

\mathbf {F} _{q}

should be mentioned only for completeness.

This is true that

\mathbb {F} _{q}

is more commonly used in pure mathematics and number theory. On the other hand,

\operatorname {GF} (q)

is more commonly used in computer science. This is mainly for historical reasons, but a practical reason exists also: in pure mathematics,

q

is generally a variable or a small integer, while in computer science,

q

is often an explicit number involving an exponent or a double exponent, which makes cumbersome to use

q

as an index: compare

\mathbb {F} _{2^{64}}

and

\operatorname {GF} (2^{64})

(this is the largest field whose elements can be represented by double machine words). I have rewritten this article some time ago, and if I have preferred the notation

\operatorname {GF} (q),

this is mainly because the notation had to be used in section headings. Otherwise, because of my personal history, I would have preferred

\mathbb {F} _{q}.

So, both notations are widely used in recent literature, and Wikipedia policy is that, in such a case, it is not to Wikipedia to establish rules. D.Lazard (talk) 21:43, 13 December 2020 (UTC)

I am inclined to agree that this is a difference between fields, rather than eras -- in my experience, GF(q) is the most common notation in the coding theory literature, for example, while it seems not to be used at all in algebraic combinatorics. --JBL (talk) 22:08, 13 December 2020 (UTC)

I agree with JBL. Having taught a wide spectrum of courses that use finite fields, I have had to use both notations, depending on the subject and text. As a finite projective geometer, I have used the

\operatorname {GF} (q)

notation almost exclusively, as that is the standard in this area. On a totally practical note, I prefer this notation since the exponents (and that is where all the action is

) are typeset in a slightly larger font, making them easier to read.--Bill Cherowitzo (talk) 23:31, 13 December 2020 (UTC)

OK, thank you all; you have convinced me to withdraw my proposal. By the way, the "Extensions" section at the end of the article uses the

\mathbf {F} _{q}

notation; should that be changed for consistency? Ebony Jackson (talk) 06:03, 16 December 2020 (UTC)

It seems that the algebraic closure of a finite field is never considered in areas denoting finite fields by

GF(q)

. So it seems better to not use this notation in this section. However I am in favour to change systematically

\mathbf {F} _{q}

into

\mathbb {F} _{q},

and to use

F

instead of

\mathbf {F}

when there is no subscript. D.Lazard (talk) 11:07, 16 December 2020 (UTC)

OK, done. Ebony Jackson (talk) 03:23, 17 December 2020 (UTC)

I have just read this. I apologize for effectively undoing the

\mathbf {F} _{q}

to

\mathbb {F} _{q}

notation changes, and will be happy to conform the notation to the suggestion again. Some questions, though:

Isn't it inconsistent if we do not simultaneously use $\mathbb {Z}$ , for instance? I assume that would be changed too.
I consider the Unicode blackboard bold characters to be problematic on some browsers, so I would stick to inline <math> as it was and not use Unicode characters. Is the mix of <math>, {{math}} not an issue, i.e. should we change everything to <math>?

—Quondum 23:58, 4 January 2021 (UTC)

Either

\mathbb {Z}

or

\mathbf {Z}

is OK for me, though I guess you are right that if we are writing

\mathbb {F} _{q}

as D.Lazard preferred, then it would make sense to write

\mathbb {Z}

too, for consistency. For other symbols other than these blackboard bold letters, however, I think it would be OK to leave them inside the math template if they look OK as they stand. I do think that that template looks a little better, for simple formulas. Ebony Jackson (talk) 01:46, 5 January 2021 (UTC)

Sorry about the parsing (now with a quick fix applied) – an early preview actually showed that construction working.

Okay, I'll wait for any other comments. They whole font/rendering thing in WP maths has been an issue for ever so long that hoping for some "perfect style" is a bit of a dream. —Quondum 02:54, 5 January 2021 (UTC)

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