Talk:Finite field

Notation

I would like to suggest using the GF(q) notation in the article, as it's easy to write GF(pⁿ), but the equivalent with F_q notation cannot be written very satisfactarily in HTML (witness the penultimate paragraph of the article). Would anyone object strongly if I changed the article to use GF(q) notation throughout? --Zundark, 2001 Dec 10

Having written that, I see that F_p^m does display correctly in Netscape. But in IE4 it looks the same as F_pm, which is not good. --Zundark, 2001 Dec 10

In IE 5.5, it displays ok but looks somewhat crappy. It seems to be safer to use the GF notation. --AxelBoldt

It seems to be time to use F_p^m as it should be supported by modern browsers. --84.178.126.105 14:07, 5 August 2006 (UTC)

155.198.17.120's changes

Maybe someone who knows about the subject wouldn't mind reviewing the changes done by 155.198.17.120 to this article. She seems to have subtly vandalised Fascism and started the bogus entry Marxianism. Thanks. --snoyes 15:15 Feb 19, 2003 (UTC)

It looks as if she changed "T^3 + T^2 + T + 1 is irreducible" to "T^3 + T^2 + T - 1 is irreducible". Though my algebra is rusty, this seems to be a good and necessary change as (T^3 + T^2 + T + 1) = (T^2+1)(T+1) whereas there are no factors (mod 3) of (T^3 + T^2 + T - 1).

Thanks. Admittedly I didn't bother trying to understand the problem (it is of course fairly easy as you have demonstrated). I just saw a sign change in the tex source and "saw red" ;-) --snoyes 15:59 Feb 19, 2003 (UTC)

how finite field may be implemented in AES cryptanalysis can be elaborated with in this article —Preceding unsigned comment added by 203.110.246.230 (talk) 11:31, 5 September 2008 (UTC)

classification section

A few things in the "classification" section bothered me, so I rewrote it. First, no effort had been made to justify the existence of an irreducible polynomial of the appropriate degree, which is by no means a priori obvious; in fact I'm pretty sure you need to prove existence of the finite field before you can assert such a polynomial exists. (Please someone correct me if this is wrong.) Secondly, after I worked on filling in more details of the existence proof, it seemed worthwhile to put the main facts of the classification upfront, and then put the derivation afterwards. Dmharvey 13:31, 4 April 2006 (UTC)

irreducible poly equation?

It seems from your examples that to actually get the multiplication table of a finite field we will need the appropriate irreductible polynomial. 1) Is there a general formula for given p and n? reference? 2) Are there simple formulas for some large class of p and n? reference? 3) If the answer to 1 is no then I think this should be mentioned--preferably with a reference.

~ interested

more on irreducible poly

I think someone who understands these things should write a paragraph or two on the construction of the needed irreducible poly. It might start out with something like

"Note that without the needed irreducible polynomial that GF(p^n) is a purely theoretical object and somewhat nebulous. There seems to be no general foumula. There are varous tables of polynomials for various p and n. Such a polynomial always exists and there are various algorithms for constructing it."

The knowledgable writer could then talk about the efficiency of such algorithms, etc. It need not be formal--just enough to impress upon the reader that you have got to have such a poly to do anything with the field

~reader

In fact, I believe that there is no polynomial-time construction of any finite field. Conjectures of the Geometric Langlands Program will give one, though. Eulerianpath (talk) 22:43, 5 January 2008 (UTC)

Finite division rings vs finite fields

I believe this removal is incorrect. Finite fields are the same as finite division rings, per Wedderburn's theorem. As such, even if the Japanese Wikipedia is different from the English Wikipedia as far what is a field/division ring, the Japanese article removed by the above edit is the perfect counterpart of our article at finite field and the link must come back. Comments? Oleg Alexandrov (talk) 18:53, 6 April 2006 (UTC)

As clarification, here is the corresponding Japanese article which DYLAN thinks should not be the interwiki counterpart of our finite field article. Oleg Alexandrov (talk) 18:58, 6 April 2006 (UTC)

I'm not saying that finite division ring is not finite field. I'm just saying that Wedderburn's theorem is about the property of finite division ring. Happening to be a finite field is just the consequence of the theorem. Therefore, it shouldn't be listed as it now is.

Same thing to Japanese interwiki DYLAN LENNON 19:15, 6 April 2006 (UTC)

Disagree with DYLAN. The result is closely enough related to finite fields to warrant a mention on this page. By the way DYLAN, thanks for expanding on your actions on this talk page; it's greatly appreciated. Please continue to do that. Dmharvey 19:45, 6 April 2006 (UTC)

But the primary thing I want to get across is not so much the removal by WAREL of the paragraph

Division rings are algebraic structures more general than fields, as they are not assumed to be necessarily commutative. Wedderburn's (little) theorem states that all finite division rings are commutative, hence finite fields.

That may be a matter of taste. My point is that since a finite field is the same as a finite division ring, the interwiki link from this finite field article must go to the Japanese "finite division ring or whatever" article, as they are talking about the same thing, even if starting with different assumptions about commutativity. Does the Japanese Wikipedia have separate entries for "finite division ring" and "finite field"? I hightly doubt. Oleg Alexandrov (talk) 20:25, 6 April 2006 (UTC)

If there is indeed only one article on wikpedia.ja that deals with this topic, I have no idea how WAREL could convince even him/herself that it improves the article to remove the link. But perhaps that wasn't really the issue. Elroch 22:30, 6 April 2006 (UTC)

Since it's a math article, we should be strictly as we can about what we are talking.　有限体 means finite division ring. Even if every finite division ring is indeed finite field, there is a difference. We won't link a artcle of "natural number" to that of "integer" even every natural number is integer, do we? DYLAN LENNON 23:50, 6 April 2006 (UTC)

Yes, actually we do. Splendid example. Dmharvey 00:23, 7 April 2006 (UTC)

DYLAN, what I am saying is that our English article about finite fields is the direct analog of the Japanese article about finite division rings. Things start with different assumptions, but the material in both articles is essentially the same (and you don't need to know Japanese to understand that). As such, the interwiki link to the Japanese article is appropriate. Oleg Alexandrov (talk) 01:13, 7 April 2006 (UTC)

Also, your example of integers is invalid. Every natural number is an integer, but not every integer is natural number. However, every finite division ring is a finite field, and every finite field is a finite division ring. Oleg Alexandrov (talk) 01:15, 7 April 2006 (UTC)

Since every finite field is a finite division ring, I am pursuaded that the link is valid. However, I think Wedderburn's theorem should not be listed here but on the article of "division ring". DYLAN LENNON 12:15, 7 April 2006 (UTC)

DYLAN, I find it amazing that you have not yet inferred that if you repeatedly make an edit with which everyone disagrees, you will get repeatedly reverted back. This has happened to you numerous times already. You will not get your way by repetition alone. Rather, you will get blocked. The only way you will get your way is if you write down an argument that convinces other editors. It is not good enough to write down an argument that convinces yourself. I find it puzzling that you haven't realised this yet, because you are obviously a reasonably intelligent person. Can you explain why you continue to change the article, when all of the evidence should make it clear that someone will revert it back straight away? Dmharvey 13:06, 7 April 2006 (UTC)

sloppy

isn't this article sloppy in discriminating between things that are equal and things that are just isomorphic? --Lunni Fring 21:11, 23 September 2006 (UTC)

I can see one place where it says 'the', and it might say 'a'. Why do you ask? Charles Matthews 21:32, 23 September 2006 (UTC)

Well one example is "Then GF(q) = GF(p)[T] / <f(T)>." But GF(q) is really isomorphic to the RHS. --149.169.52.67 22:37, 23 September 2006 (UTC)

Lead Section Inconsistency

The definition "GF is a field that contains only finitely many elements." is inconsistent with the fact that a field of 6 elements has a finite number of elements, yet it is not a GF. John 20:12, 13 March 2007 (UTC)

There is no field of six elements, so there is no inconsistency. --Zundark 22:36, 13 March 2007 (UTC)

It seems to me that is true only if you define a field as a GF, and that makes it a circular definition. Why is there no field of 6 elements? Do we need to clarify or define field so that is true? Otherwise why should a reader who is reading this to find out what a GF is assume that? John 05:47, 15 March 2007 (UTC)

This article links to the field (mathematics) article, which provides the definition of field. The non-existence of fields of 6 elements follows from the definition by elementary arguments (which are already given in the article). --Zundark 09:45, 15 March 2007 (UTC)

You are right, I agree, I was under the mistaken impression that field was not defined or linked. I see now. Thanks. John 16:54, 15 March 2007 (UTC)

Sum of all field elements

Please see Talk:Field (mathematics)#Sum of all field elements. —The preceding unsigned comment was added by 80.178.241.65 (talk) 20:27, 15 April 2007 (UTC).

Naming of "Galois Fields"

I have been told that referring to finite fields as "Galois fields" is a little odd, with the reason being that Galois himself never worked with finite fields. This may be worth mentioning if anyone can confirm it. 81.182.122.2 19:11, 15 May 2007 (UTC)

The EoM entry on Galois fields says they were first considered by Galois, and gives a reference. --Zundark 20:05, 15 May 2007 (UTC)

Galois not only worked with finite fields, it's quite probable that he was the first one to come up with the concept. See Évariste Galois, "Sur la théorie des nombres". Bulletin des Sciences mathématiques XIII: p. 428ff (1830). It's also part of the collection of Galois' mathematical works published by Joseph Liouville in 1846, and the Bibliothéque Nationale de France helpfully makes this available here (use the interface to go to page 381. It's in French, of course, but those are the breaks). --Stormwyrm (talk) 04:45, 19 February 2009 (UTC)

Proof that a Finite Field has Order [tex]p^n[/tex]

The first proof claims that a finite field is a vector space over Z/pZ, where p is the characteristic, and then goes on to conclude that the order of the field must be [tex]p^n[/tex]. But it seems proving that it is a vector space over Z/pZ is the heart of the proof; why leave it out? —Preceding unsigned comment added by 76.167.241.175 (talk) 07:08, 25 February 2008 (UTC)

simpler explanation

Something I think would be useful is to be told that the additive group of a finite field is primitive abelian, and its multiplicative group is cyclic. This seems to me to explain in simple terms exactly how every finite field works. But instead I see stuff like "(Z/2Z)[T]/(T²+T+1)", which is completely opaque to me, and I suspect far more complicated than necessary. Maproom (talk) 08:56, 29 May 2008 (UTC)

The article already mentions that the multiplicative group is cyclic. What does primitive abelian mean? In any case, information about the additive and multiplicative groups does not "explain exactly how every finite field works", since it is the relationship between them that matters. Algebraist 16:03, 11 February 2009 (UTC)

A "primitive Abelian" group is the direct product of a number of cyclic groups of prime order. Maproom (talk) 16:08, 11 February 2009 (UTC)

That's mentioned in the article too. Do you think it should be more prominent? Algebraist 16:12, 11 February 2009 (UTC)

The additive group of a finite field is an elementary abelian group. There may be a translation problem; there is exactly one hit from mathscinet for "primitive abelian group", and such a group is torsion-free, so never finite. A primitive permutation group that is abelian is cyclic of prime order, and this appears to account for all other usages on google.

At any rate, to agree with both Maproom and Algebraist, viewing a finite field as an especially nice subring of the full matrix ring M_n(Z/pZ) is a very good way to understand finite fields that explicitly describes the additive group, the multiplicative group, and the action of the multiplicative group on the additive group. A reasonable choice of subring is the polynomials in the companion matrix of an irreducible polynomial of degree n over Z/pZ. This also explains many applications such as LFSRs. JackSchmidt (talk) 16:48, 11 February 2009 (UTC)

Reference suggestion

I suggest that reference be made to "Arithmetic over finite fields" in Collected Algorithms of ACM (?467) by Lam & McKay. It uses Zech logarithms, is short and efficient. John McKay132.205.221.161 (talk) 15:53, 12 February 2009 (UTC)

the finite field is not completely known

I tried to delete the following passage "The finite fields are completely known", first of all because it is a false statement, secondly, since when is something "completely known"? There is no reason to emphasize that a theorem completely holds, and thirdly and most important the quoted sentence imply that there is nothing new to discover in the area of finite fields and that research in that area is pointless. Whatever you (the nice editor that reverted my delete) are trying to say it comes off wrong.

I think the wording should change to include a context, or be deleted altogether, please express your concerns if you have them or I'll delete it again. —Preceding unsigned comment added by Sonoluminesence (talk • contribs) 09:03, 28 March 2009 (UTC)

I (the reverter) am not trying to say anything. An unexplained deletion of a large percentage of the introduction is a standard thing to revert (indistinguishable from random vandalism). An WP:edit summary is normally sufficient to explain the deletion, but in this case it is better to try improve the introduction.

We can guess at what the original editor meant, and try to improve it. A purpose of the WP:lead section, amongst others, is to summarize the article, and at least half the article is describing the classification of finite fields. Presumably the original author was trying to summarize this. I changed the wording to be more direct, "The finite fields are classified by order." JackSchmidt (talk) 05:58, 29 March 2009 (UTC)

Hello JackSchmidt, thank you for your response and change, I think the wording is now exponentially better. Sonoluminesence (talk) 07:48, 29 March 2009 (UTC)

"The finite fields are fully known" makes sense to me, and tells me something useful about finite fields. "The finite fields are classified by order" merely tells me how people have chosen to classify them. The former statement (if true, which I think it is) seems far more useful. Maproom (talk) 22:24, 29 March 2009 (UTC)

Unfortunately there are many very basic questions about Z/pZ that are not known, so the statement is either meaningless (my opinion) or false (probably Sono's opinion). A silly example of great practical importance is implicit in Primitive root modulo n#Order of magnitude of primitive roots; without the GRH estimates on the smallest generator are fairly poor, so until GRH is proven, our knowledge on even a very basic question is extremely incomplete (and even then, I don't know that the estimates are sharp enough for more serious computational applications). At any rate, if you can find WP:reliable sources to support the claim, then feel free to expand the article (not the lead) with the new material. This will make a good excuse to fix the current introduction which neglects to summarize over half the article. JackSchmidt (talk) 22:43, 29 March 2009 (UTC)

Also, to make it clear research is on-going in finite fields (which might make a nice "Finite fields are completely classified, but are still actively researched." sort of comparison), there is the short section Kakeya conjecture#Kakeya sets in vector spaces over finite fields. There are lots of problems over uncountable fields that are not resolved over finite fields (and lots of the reverse). Basically finite fields are a healthy, vibrant area of research mathematics. JackSchmidt (talk) 22:56, 29 March 2009 (UTC)

I'll be happy with "Finite fields are completely classified, but are still actively researched." "Completely classified" is the kind of thing I can understand, and want to know. Maproom (talk) 08:49, 30 March 2009 (UTC)

Why do you insist on putting the word "complete"? It doesn't appear with that context in the article. If you think that it is complete, please explain in what way are the classifications complete? The wording you suggest doesn't contribute anything - not about the finite field nor its classifications, the current wording actually says something new which is explained right on the next paragraph. I vote to keep the current wording. Sonoluminesence (talk) 10:55, 31 March 2009 (UTC)