Talk:Field (mathematics)

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== The definition should be changed ==well what they are talking about is really based on math and science and how it helps alot of people and how if it wasn't for mathematics then we would'nt know what certain things are but now that we fot mathematics we don't have to worry about it because we know what it is and how mathematics works.....

Thank You Sincerly.Ms.Wiliams

The definitions of group (semigroup), ring (semiring), fields are really confusing. I think that the following definition is much clearer. My idea is to compare those concepts together. If you agree, please help to put them right. If you find any problem, please let me know. Jackzhp 22:43, 4 September 2006 (UTC)

F is a set
Two binary operators are generally marked as '+' and '.', but you can use any symbols you like. When F is a set of subsets of another set, the two operators are generally marked with '∩' and 'U'.

A field (F, two operators) satisfies the following property:

1. F is closed for both operations defined by the two binary operators.
2. One operation is distributive over the other.
3. both operations are associative.
4. Both operations are commutative.
5. Identities exist for both operations.
6. Inverses exist for both operations.

For reference, group, semiring, ring are defined as follows: A group (F,one operator) satisfies the following property:

F is closed for the operation.
The operation is associative.
Identity for the operation is included.
Inverses for the operation is included.

A semigroup doesn't require the identity and inverses.

A ring (F, two operators) satisfies:

1. F is closed for two operations.
2. One operation is distributive over the other.
3. Both operation are associative.
4. One operation is commutative.
5. The commutative operation has identity and inverse.
6. The other operation has identity,

A semiring doesn't require the inverses for the commutative operation included in F.

A ring becomes a field when two operations are commutative and have inverses included.

A field is a ring, a ring is a semiring, a semiring is a group.

Contents

[edit] Nimbers? Birthdays?

What are nimbers?

Since when do numbers have birthdays?

—Preceding unsigned comment added by 137.111.132.xxx (talk • contribs) 23:30, August 21, 2001

Surreal numbers have birthdays. 0 ({|}) was born on day 0, 1 ({0|}) and -1 ({|0}) were born

on day 1. 2, -2, 1/2, -1/2 were born on day 2. Etc. See surreal number. —Preceding unsigned comment added by Luqui (talk • contribs) 05:03, October 16, 2004

On this topic, is it true that only the sets of surreals with birthdays less than inaccessible cardinals form fields? As far as I can tell those with birthdays less than omega^omega should form a field, but I might be mistaken...--4.245.82.95 01:56, 20 Mar 2005 (UTC)

[edit] "Could someone …"

Could someone please illustrate the orders that make the examples given at the end into fields?

[edit] help?

Just edited a few pages linking here to link to field (mathematics), but there are much more left.

Can someone please help me? --JensMueller 19:35, September 6, 2003 (UTC)

[edit] TRACE?

Could someone better than me please add a page describing what the TRACE of an element of a field is? —Preceding unsigned comment added by 80.129.231.225 (talk) 00:09, December 8, 2004

There's an article at field trace if that's what you are looking for. -- Fropuff 01:53, 2004 Dec 8 (UTC)

[edit] Conmutative property is not needed

As far as I recall the conmutative propierty is not included in the definition of a field. A field withs such propierty is called a conmutative field. Though most fields are conmutatives, there are some wich aren't (only quaternions come to mind right now). Could someone confirm and edit the page accordingly? —Preceding unsigned comment added by 81.202.172.209 (talk) 15:35, February 25, 2005

Definitions vary, but I think the most common definition of a field is a commutative one. The more general definiton you refer to is called a division ring. -- Fropuff 17:07, 2005 Feb 25 (UTC)

So all authors of papers corresponding to the > 3500 google hits for

"commutative field" -"non-commutative field theory" -"commutative field theories"

(i.e. all relating to NCFT is excluded!) are not up to date...

Noncommutative field theories come from physics and have nothing to do with this page.

The usual terms for the noncommutative case are (a) division rings (b) skewfields (c) skew fields as well as on occasion (d) noncommutative fields. But fields as such are commutative. Abu Amaal 20:36, 24 February 2006 (UTC)

I can't see why those not working on the subject should know better than those who do work on the subject. — MFH: Talk 15:34, 27 May 2005 (UTC)

It's not a question of up-to-dateness, it's a question of what is the most common usage. I don't think there's much doubt that fields are usually defined to be commutative. The Google results really don't mean much - Google claims 6880 hits for "associative ring", but that doesn't mean we should remove associativity from our definition of ring. --Zundark 21:11, 27 May 2005 (UTC)
From somebody not working in the area. :) Yes, fields are by default assumed commutative. I am not sure if for a noncommutative field K one can still talk about the polynomials K[X] being an Euclidean domain or its ring of fractions. So, in addition, the commutative case is the most important case anyway. :) Oleg Alexandrov 23:54, 27 May 2005 (UTC)

[edit] Definition, should not depend on ring?

Would it not be an idea to provide a definition for a field that does not rely upon the definition for a ring? This is a good idea for many reasons, but first of all, because the wikipedia definition for a ring is riddled with errors and discussions, and seems to change on a daily basis. Secondly, because fields are much easier to comprehend, and the study of fields generally does not rely on the prior knowledge or study of rings. —Preceding unsigned comment added by 213.187.172.150 (talk) 14:05, June 5, 2005

  1. Unfortunately, the theory describes a field in either one of two ways: 1) As a commutative ring... and 2) As a Set with these properties... But theoretically both statements are equally true. But the definition using a ring is more used because a field must be a ring before being a field. Second, because it's easier to enlist the additional properties that a ring must have in order to be a field. Maybe you could help by correcting the Ring article...Andycyca|say... 03:14, 22 August 2006 (UTC)

[edit] Why is a field called a field?

Can someone enlighten me as to why a field is called a field? Xiaodai 09:35, 17 August 2005 (UTC)

See http://members.aol.com/jeff570/f.html and then http://mathforum.org/kb/message.jspa?messageID=1186458&tstart=0
The Germans used "Koerper" then as now but at the time "Bereich" was in competition. Abu Amaal 22:23, 24 February 2006 (UTC)

[edit] grimmett and stirzaker's dfn.

Grimmett and Stirzaker define a field as having (1) the empty set (2) the union of any elements of the field is also an element of the field and (3) every elements compliment must also be in the field. Are they crazy, just going after something else, or secretly much smarter that other mathematicians --Pdbailey 03:59, 22 September 2005 (UTC)

This is a field of sets. --Zundark 21:01, 24 February 2006 (UTC)

[edit] Integral Domains

This page ought to mention integral domains. Maybe that is thoroughly dealt with elsewhere, but the term should occur here with a link to the relevant page(s). The point is that every integral domain has a field of fractions and this is an important source of examples of fields. From the integers to the rationals, from polynomials to rational functions, etc.

The list of examples seems a bit diffuse. The core examples are number fields, function fields, finite fields, their completions (including Laurent series fields) and their algebraic closures. Abu Amaal 22:13, 24 February 2006 (UTC)

[edit] Japanese interwiki

Though I'm not inclined to trust him on the point, I also can't be sure WAREL is wrong about the Japanese article referring to skew fields. Could someone who reads Japanese please check this out? --Trovatore 19:27, 4 April 2006 (UTC)

The Japanese article ja:体 (数学) uses the definition for skew fields, but it also mentions the commutativity condition:
さらにもう一つ、乗法の可換性に関する条件
  • K のどんな元 a, b についても、 ab = ba が満たされる。
を加えるとき可換体と呼び、可換性が満たされない元を持つとき非可換体と呼ぶ。
and then it goes on to explain the difference between the English-speaking world and continental Europe. Therefore, both field (mathematics) and division ring should have a link to ja:体 (数学). See also this diff. -- Jitse Niesen (talk) 05:10, 5 April 2006 (UTC)

It does explain that ,in English, field always means its commutative. However, it states that they will use the term "体(field)"　as division ring through the entire Japanese wikipedia and they actually are using it as so. WAREL 05:20, 5 April 2006 (UTC)

As far as I can tell the only article in the Japanese wiki which is about fields (i.e. commutative division rings) is ja:体 (数学). WAREL is there a different article devoted exclusively to commutative division rings? If not then this article should interwiki link to ja:体 (数学). Paul August ☎ 05:55, 5 April 2006 (UTC)
So, the Japanese article ja:体 (数学) is about division rings, it mentions fields (in fact, all but one of the examples are fields, and a lot of the next section (諸概念) also seems to be valid for fields), and there is no article in Japanese which is purely about fields (which I guess would be called something like 可換体 in Japanese?). If this is correct, I agree with Paul that an interlanguage link to ja:体 (数学) would be useful to the reader. Unfortunately, I'm having trouble finding the policy regarding interlanguage links. -- Jitse Niesen (talk) 06:05, 5 April 2006 (UTC) (via edit conflict)
i did write on ja:体 (数学) and ja:体の拡大 which deals about field extensions, i think, using 可換体 for fields is a especially-speaking, but usually they say simply 体 and use 斜体 or 多元体 for divisions (using 体 for divisions is of naive-dealing or for compound term). so, I also believe it would be useful to interlang-link to (and linked from)ja:体 (数学). if possible, then make article ja:多元体 which links from division ring to. --218.42.231.45 14:07, 5 April 2006 (UTC)

Yes, 可換体 is the term for field. Some Japanese books use 体 for field, but 可換体 is used for field and 体 is used for skew field through the entire Japanese wikipedia articles.WAREL 16:47, 5 April 2006 (UTC)

Thanks for your reply WAREL. Is there a seperate ja article on 可換体? I can't find it. Paul August ☎ 18:16, 5 April 2006 (UTC)
When consensus is reached, someone else can remove the interwiki, WAREL. -lethe talk + 17:10, 5 April 2006 (UTC)
WAREL, your opinion is too non-sense. 体 means "field" or "skew field", depending on their lying context. And prefixes 可換- and 斜- (or 非可換-) is used for disambiguation. This does not mean your "体 is used for skew field through the entire Japanese wikipedia articles". / 日本語版が書いてるのは、明確化のために field を可換体、skew field を非可換体と接頭辞をつけて呼び分けることがあるということにすぎないので、「field は可換体、体は skew field というように日本語版は書いている」というあなたの主張は、日本語を読める人からすればちゃんちゃらおかしいですよ。馬鹿を言うのも休み休みにしてください。--218.251.73.166 00:42, 6 April 2006 (UTC)

Through the whole Japanese wikipedia articles, 体 totally means "skew field". Give me any counterexample.DYLAN LENNON 18:18, 6 April 2006 (UTC)

just one of simple counterexample for you is the very ja:体 (数学). --218.251.72.223 15:32, 10 April 2006 (UTC)

By the way, there is no article of 可換体 in Japanese wikipedia.DYLAN LENNON 18:34, 6 April 2006 (UTC)

DYLAN, thanks for answering my question. Since that is the case, this article should link to ja:体 (数学), since it is the closest equivalent Japanese article, don't you agree? Paul August ☎ 19:59, 6 April 2006 (UTC)

I disagree. 体　sometimes means "field" depending on the context. In the context of Japanese articles, this link only causes misunderstanding. DYLAN LENNON 12:04, 7 April 2006 (UTC)

Misunderstanding? You should use such a word after you understand... --Schildt.a 15:59, 7 April 2006 (UTC)

After all, what I said is right. Schildt is not good at English or at Japanese. 218.133.184.53 05:32, 6 June 2006 (UTC)


[edit] Meaning of Körper

In the present text, the reader might get the impression that German Körper and Spanish cuerpo literally mean "field", which is false. I guess the intention is to convey that in mathematical use these terms mean the same as the mathematical term "field" in English; in particular that commutatitivity of multiplication is implied. Perhaps some-one who is sufficiently familiar with both German and Spanish mathematical terminology to confirm this could adjust the text so that no misunderstanding can arise. LambiamTalk 20:08, 4 April 2006 (UTC)

An algebraic field is Körper in German, even though it literally means "body". - grubber 05:02, 5 April 2006 (UTC)
Right. However, the issue I raise is that the text of the article at this point is ambiguous, that is, amenable to different interpretations, giving the risk of a false impression an average not particularly mathematically-schooled reader who doesn't know German or Spanish too well, and who probably is not going to read this talk page, might take away from the text of the article. LambiamTalk 23:02, 5 April 2006 (UTC)
Good point. I reformulated the paragraph in an attempt to clear it up. However, I'm starting to think there might be a bit too much about other languages. I can see why French (Galois) and German (use of K for fields) are mentioned, but we should be careful not to discuss the usage in every language on earth. -- Jitse Niesen (talk) 00:42, 6 April 2006 (UTC)
An anonymous edit just added the Polish version of the word. I agree tho, German and French are justifiable, but I'm not sure why others should be mentioned. I mean, if you want to know the word in another non-notable language, then you can click the interwikilink. I'm tempted to remove all but the French and German. - grubber 20:22, 11 April 2006 (UTC)
Polish is weird, since Polish has never been a lingua franca of mathematics, unlike French and German. -lethe talk + 20:29, 11 April 2006 (UTC)
I removed the Spanish, Italian and Polish words from the article. -- Jitse Niesen (talk)
I agree. At most they belong in Wiktionary. Melchoir 07:08, 13 April 2006 (UTC)
As fas as I can tell Körper is not exactly a field, it is a noncommutative field 132.239.145.119 22:49, 22 January 2007 (UTC)
No, Körper is field. A "noncommutative field" is called a "division ring", which is Schiefkörper or even Divisionsring. - grubber 23:55, 22 January 2007 (UTC)

[edit] Characteristic 2

I have a curious question: Think about a different definition of addition over the field of nonnegative numbers: the "sum" of two nonnegative numbers is defined as the absolute value of the difference of the two numbers, and the "product" is the same as ordinary multiplication. As far as I can tell, this new field satisfies all the field axioms ("multiplication" distributes over "addition", both operations are closed, commutative, and associative) and has the additional property that the additive inverse of any number is itself. This would seem to imply that it is a field of characteristic 2. However, in this field,

(x+y)^2 is not equal to x^2 + y^2.

Why is this? CecilBlade

The addition isn't associative. Melchoir 21:00, 21 April 2006 (UTC)
Are you sure? For every triple of numbers I've checked out it is. CecilBlade 21:50, 21 April 2006 (UTC)
Yeah, you need three numbers that are different and nonzero. 1+2+3, for example. Melchoir 21:59, 21 April 2006 (UTC)
Oh, thanks a lot. I can't believe I missed that! 24.14.162.255 02:04, 22 April 2006 (UTC)
It is well-known that (a+b)^n = a^n+b^n if n is a prime number equal to the ring's characteristic (algebra) — MFH:Talk 19:52, 2 June 2006 (UTC)

[edit] Japanese

(I'll use the Doonesbury convention of putting English words in angle brackets to represent foreign words.) It is a bit of an annoyance that both field (mathematics) and division ring point to the same article, <field>, but I don't see that there's any alternative as long as the ja.wikians see fit to keep the two concepts in a single article. Would anyone be interested in asking them if they think it's a good idea to divide the articles? Since most of the interesting information is about (commutative) fields, the way I'd do it, if I were they, is first move <field> to <commutative field>, making appropriate changes, and then write a new <field> article corresponding to our division ring. That this would straighten out our interwiki problems is presumably a minor consideration to them, but it's a change that could conceivably make sense for its own sake (obviously, assuming they think so, without any interference from WAREL socks).

Now of course I don't think this would solve our problems with WAREL, who (I predict) will just move on to something else. But it could be an improvement on its own merits. --Trovatore 16:47, 16 May 2006 (UTC)

Oh, just for completeness, not really advocating it, another possibility would be for us to merge field (mathematics) and division ring into a new article, perhaps called fields and division rings. Don't really like it because there's no good name for it (in principle, of course, we could just call it "division ring", but the problem is that the more interesting stuff is about fields). Just thought I'd mention that there's another possibility than expecting everyone else to have a parallel article strucure to ours. --Trovatore 19:07, 16 May 2006 (UTC)

[edit] Integers Mod 2?

Would the integers mod 2 be a field, it has after all for any element a != 0 in Mod2 another element b in Mod2 such that a+b/ab can either equal the additive or multiplicative idenity respectively? Is this why computers have addapted a binary system? ~Richard Detsch —The preceding unsigned comment was added by 72.71.209.132 (talk • contribs) 16:07, 2 June 2006 (UTC)

They sure are! See Finite field. Melchoir 17:05, 2 June 2006 (UTC)

[edit] Definitions

It's important to start out with a jargon-free definition in the first sentence and to present multiple, equivalent, precise formulations below. Please don't wipe out this material. Melchoir 22:23, 11 November 2006 (UTC)

I don't think this change: [1] is an improvement]. I would suggest leaving the previous paragraph as an introduction, but perhaps adding: Fields are a special case of rings, but differ from them in that division is required to be possible in fields, but not necessarily in rings. The prototypical example of a field is Q, the field of rational number]s. Joeldl 17:37, 22 March 2007 (UTC)

Go for it! Septentrionalis PMAnderson 17:52, 22 March 2007 (UTC)

The notation Z/pZ has recently been changed to GF(p). I am of the opinion that the first is more common, although Fp is certainly not uncommon. Also, I think that because people are more familiar with modular arithmetic than they are with fields of cardinal pa, a≥2, it's best to connect Z/pZ with modular arithmetic rather than finite fields. Joeldl 15:02, 23 March 2007 (UTC)

I agree that Z/pZ is preferred, the GF(p) notation looks kind of obscure and I think should not be used except in rather specialized contexts. Oleg Alexandrov (talk) 15:05, 23 March 2007 (UTC)

[edit] Sum of all field elements

Is it true that the sum of all the elements of a given finite field of more than 2 elements must be 0?

How can it be proven? 80.178.241.65 20:02, 15 April 2007 (UTC)

Hm, let me take a stab at this. Let the field have size p^n. We know that the sum of the numbers S=1+2+...+p^n-1 = (p^n-1)*(p^n)/2. If p is odd, then (p^n-1) is even and p^n divides S. If p=2, then 2 divides S only if n>1. So as long as p^n>2, p divides S. Every finite field is cyclic, so let a be a generator for the field. Add all the elements of the field together: 0+a+2a+3a+...+(p^n-1)a = a*S = 0 mod p. - grubber 00:09, 16 April 2007 (UTC)
Every finite field is not cyclic, for example the field with 4 elements is not Z/4Z, which is not a field. Here is a solution, though. Let k be the field. Pair together all elements x with their opposites -x. There will usually be two elements here. The only exception is when x = -x, that is, 2x=0. If char k is not 2, then this only happens for x = 0, so it is true. If char k = 2, Then this argument falls apart because every element is its own opposite. But view k as a finite dimensional vector space over Z/2Z, say of dimension n, and choose a basis. Then the coordinates of the sum are obtained by adding up the coordinates of all the vectors, i.e. all the elements of k. There are 2n-1 whose first coordinate is 0, and the same number for which it is 1. If n ≥ 2, then 2n-1 is even, so adding them up in Z/2Z gives 0. Do the same for all coordinates. The sum is 0 in each coordinate. So yes, the only exception is when k has 2 elements. Joeldl 08:35, 16 April 2007 (UTC)
It's been a while since I looked at field theory, so you're right. Big mistake on my side -- sorry. I was thinking of primitive elements, which says the multiplicative group is cyclic, but not cyclic in any way that is related to the additive group (except for fields of prime order). Thanks for the correction! - grubber 13:24, 16 April 2007 (UTC)

[edit] Requirement "0 and 1 are different" is not convenient.

If the field has two elements or more, then the requirement 0\neq 1 is useless, because it is a corollary of the other conditions (0 * a = 0 and 1 * a = a with an a such that 0\neq a). On the other hand, without trivial (singleton) fields one cannot say that a concrete trivial vector space is a universal algebra defined by a set of operations, where the unary ones correspond to the underlying field in a 1-1 way.

The case of a trivial vector space that is a (universal) subalgebra of a larger space does not deny this statement, because the universal subalgebra construction does not require neither the preservation of such underlying fields nor in general the "preservation" of the set of operations (in the sense that the restricted operations cannot coalesce). There is a preservation only for universal algebras defined by an indexing of operations, which might be not necessary for vector spaces. Why to deny vector spaces the former definition?

Therefore, allowing 0 = 1 saves some ink and makes Linear Algebra closer to Universal Algebra. --Gabriele ricci (talk) 17:15, 6 April 2008 (UTC)

I think we should follow existing conventions. From what I know, the set {0} is not considered a field, the smallest field is {0, 1}. Oleg Alexandrov (talk) 17:45, 6 April 2008 (UTC)

Not all existing conventions conform to the requirement 0\neq 1. Many of them do, when they use groups through rings as in definition 1 or 2. (we could keep them by merely adding the restriction that they hold for non singleton fields.)On the contrary, when we are not stressing groups as done in definition 3, we can simplify the definition and make it more general. This is what some (non group addicts) authors do, e.g. at p. 23 of R.R. Stoll, Linear Algebra and Matrix Theory, McGrow Hill (1952). Likely, adding a footnote about the "0 = 1" view might make everybody happy. --Gabriele ricci (talk) 15:27, 9 April 2008 (UTC)