Talk:Field (mathematics)/Archive 2

merger proposal

I suggest merging field (mathematics) and field theory (mathematics) until the articles have enough content to justify a separation. (Such as it is the case with group (mathematics) and group theory, for example). Anybody against? Jakob.scholbach (talk) 12:50, 1 August 2008 (UTC)

The merge was undone on the field theory side. The editor left no explanation, but their edits generally seems sane. JackSchmidt (talk) 14:29, 15 August 2009 (UTC)

This is linked to by psych/soc stats, so I think that it should remain.--John Bessa (talk) 15:54, 18 May 2011 (UTC)

to do list

Here are some points I wish to include in the article. If others are up to it before I do, go ahead!

• the field of constructible numbers & non-constructibility of 3rd square root of 2, mention classical Greek problems
• algebraic numbers/transcendental should be a central organizing theme of the article, examples π, transcendence of π, open questions in this direction
• main theorem of Galois theory, mention infinite case
• Lefschetz principle, ${\overline {\mathbb {Q} _{\ell }}}\cong \mathbb {C}$
• Applications? Jakob.scholbach (talk) 13:40, 8 August 2008 (UTC)

I added a section on Galois theory as a starting point. Feel free to edit it. RobHar (talk) 18:09, 8 August 2008 (UTC)

Good. I'll be back in 2 weeks. Looking forward to working on this stuff. In general, the material seems already promising, but restructuring and so on will be necessary. Jakob.scholbach (talk) 19:15, 8 August 2008 (UTC)

Typo?

I'm not a mathematician, but the following statement struck me as odd ("Definition and Illustration" section 4):

Additive and multiplicative identity: There exists an element 0 ∈ F, called additive identity element, such that for all a, a + 0 = a. Likewise there is an element 1, the multiplicative identity element, such that for all a belonging to F, a · 1 = a. For technical reasons, 1 is required not to equal 0.

Should the last sentence not be:

For technical reasons, a is required not to equal 0.

To me, it's obvious that 1 is not 0 :-)

--209.202.71.82

No, it's correct as it is: we want 0 and 1 to be different in every field. --Zundark (talk) 16:41, 4 September 2008 (UTC)

a · 1 = a must hold even for a = 0. And if you drop the requirement that the additive identity (denoted by 0) and the multiplicative identity (denoted by 1) be distinct, the ring with one element becomes a field. We want to have a field with one element, but not that one (see http://math.ucr.edu/home/baez/week259.html). --A r m y 1 9 8 7 ! ! ! 12:17, 6 September 2008 (UTC)

I think the person's question more revolved around the statement $0\neq 1$, and is confused because s/he is accustomed to seeing 0,1 as integers. Here, 0 and 1 are simply names representing elements from the set. We might as well call them $a_{0},a_{1}$. The constraint $0\neq 1$ is implicit if you think of 0, 1 as integers, but is not for $a_{0}\neq a_{1}$, and so we must mention the constraint explicitly in this context.

I've clarified it a little. -- Army1987 (t — c) 14:44, 26 October 2008 (UTC)

89.7.98.15 (talk) 11:21, 9 April 2009 (UTC)

I would like to report the following text (copied from the Generalizations section):

There are also proper classes with field structure, which are sometimes called Fields, with a capital F:

The surreal numbers form a Field containing the reals, and would be a field except for the fact that they are a proper class, not a set. The set of all surreal numbers with birthday smaller than some inaccessible cardinal form a field. The nimbers form a Field. The set of nimbers with birthday smaller than 22n, the nimbers with birthday smaller than any infinite cardinal are all examples of fields. In a different direction, differential fields are fields equipped with a derivation. For example, the field R(X), together with the standard derivative of polynomials forms a differential field. These fields are central to differential Galois theory. Exponential fields, meanwhile, are fields equipped with an exponential function that provides a homomorphism between the additive and multiplicative groups within the field. The usual exponential function makes the real and complex numbers exponential fields, denoted Rexp and Cexp respectively.

Generalizing in a more categorical direction yields the field with one element and related objects.

I have posted this error here as I wasn't able to find any other place to do so.

89.7.98.15 (talk) 11:21, 9 April 2009 (UTC)

Which error are you talking about? What is your point? — Emil J. 11:27, 9 April 2009 (UTC)

As an idea, most books I've seen either use letters for the identities (eg. e), or subscript the symbol for the field at the end (I don't know how to do it in wiki latex thing, but in normal LaTeX I mean something like 1_F, 0_F) to emphasise that these are the multiplicative and additive identities with respect to a field. What is there is fine of course, but maybe that's a way to clear it up (and then a sentence later saying that when the context is understood, 0 and 1 are used or something. 203.214.120.215 (talk) 03:28, 9 May 2009 (UTC)

Partial or Total Order?

Is there any notion of a partial or total order for fields? Is there any well-studied structured set constrained to include total or partial orders?

Or in other words,

If I needed to give a name to a set with + (associative,commutative), - (associative), * (associative, commutative), /, and <= (either partial or total), what would I call it? —Preceding unsigned comment added by 140.180.54.117 (talk) 03:20, 26 October 2008 (UTC)

There is a thing called an ordered field. That's a field equipped with a total order compatible with the field operations. I've never seen anything about fields equipped with partial orders. Algebraist 03:45, 26 October 2008 (UTC)

Thanks! The reason I ask is that I am writing software to represent polytopes in spaces with at least one cyclic dimension. I am using convex hulls to represent these polytopes (though a half-space representation would yield the same problems). For instance, imagine two convex polygons $A,B$ on the surface of a torus. Unless I'm mistaken, I don't believe I can compute $A\cap B$ without at least a partial order. I am exploring a technique of lifting each dimension of the space into the reals, and then lowering it back down after the computation (this is not the prettiest solution). Still, I hope there is some well-studied mathematical structure that fits my purposes better. —Preceding unsigned comment added by 140.180.54.117 (talk) 13:59, 26 October 2008 (UTC)

Question

The number 1 is considered the "identity element" for multiplication. In other words any number, when multiplied by the identity element, yields itself. The number 0 has a unique function also, i.e. that any number, when multiplied by zero, yields zero. Granted, the article explains how this can be proven from other properties, yet I have the distinct impression that this behavior that the number zero exhibits with respect to multiplication also has a name, which unfortunately I cannot remember at this moment. I believe that this name ought to be in these articles on algebras, and it is not. I would like to see that terminology added, whatever it is. —Preceding unsigned comment added by 72.68.150.155 (talk) 01:09, 16 December 2008 (UTC)

Z (integer) not defined or linked

As a user, trying to understand finite fields, etc. I noticed that Z is not defined or linked, even though R, C, etc. are. I recommend making the first occurrence of Z link to Integer. (Note that searching for Z requires figuring out which item in the Disambiguation list is appropriate.) Grossrider (talk) 01:48, 23 February 2009 (UTC)

Done. Feel free to do this kind of things right away ... Jakob.scholbach (talk) 08:43, 23 February 2009 (UTC)

Etymology?

This sentence needs clarification:

In 1871, Richard Dedekind called a set of real or complex numbers which is closed under the four arithmetic operations a "field". For this notion he used the German word Körper, which means "body" or "corpus" (to suggest an organically closed entity), hence the common use of the letter K to denote a field.

That isn't really clear. So did he call it a "field" or a "Körper" (I assume the latter being a German)? And if the latter means "body" or "corpus", how do you get field from that? I'd love to see it clarified. —Preceding unsigned comment added by 130.233.178.165 (talk) 13:23, 8 October 2009 (UTC)

You are right, saying that the quoted sentence is not clear. In fact, German mathematicians, including Dedekind, don't / didn't use the word "Feld" (German for "field") for the algebr. structure discussed here. (For the other meanings of "field" in science, Germans use "Feld".) Same is true, BTW, in French. Instead, Germans say "Körper" (and French "corps"). There is no etymology from "Körper" to "field". I don't know why this word was introduced (for our alg. struct.) in English, nor who did it and when. May-be that the translations given above where considered inacceptable, so one had to find something ...--UKe-CH (talk) 22:17, 8 October 2009 (UTC) In fact, I saw now that the who and when for the English name is given soon after the sentence we discuss here ;-) --UKe-CH (talk) 22:42, 8 October 2009 (UTC)

Interwiki link to French

This link goes to "Corps (mathématiques)". This is not an exact equivalent to "field", as it is used in French wikipedia (following Bourbaki) for any division ring. An equivalent would be "corps commutatif" for which there is no French article (there being no need for one). Any suggestions?--UKe-CH (talk) 22:24, 8 October 2009 (UTC)

Interwiki links are not necessarily 1-to-1. The topic of fields (qua commutative division rings) is covered in the French article on division rings, hence it is appropriate to link to said article. — Emil J. 09:58, 9 October 2009 (UTC)

Non-Field; A counterexample?

If I'm understanding this correctly, and I'm not sure that I am, the integers would not be a field because the multiplicative inverse would not be a member of the set. For example, the multiplicative inverse of 2 is one half. 2 is a member of the set of integers, but one half is not.

If this is correct, I think it might be helpful to add something about this to the article. In other words, explain what a field is not only by giving examples of fields, but also by contrasting with an example of something that is not a field. --DavidConrad (talk) 14:01, 9 October 2009 (UTC)

For the (set of) integers not being a field (with usual addition and multiplication) and your argument for this, you are right.

The "corresponding" articles in French and German give the example of the (ring of) integers as examples of non-fields. So the same might be made here. But to pretend that this is necessary would tend to imply that for each field axiom, a counter-example that doesn't satisfy the axiom should be given (this is only possible as far as the axioms are independent, which probably isn't the case, because it is generally believed to be unpractical and/or not so good for understanding). So, for instance, an example of a division ring that isn't a field would also be needed - this example would be the div. r. of quaternions (to keep it as simple as possible). Etc.: clearly this would go too far. So, such counter-examples should only be given, when they are needed to avoid common mistakes or to help understand part of the definition. I don't think this is the case here.--UKe-CH (talk) 15:22, 12 October 2009 (UTC)

I agree. I looked for a good place to put the nonexample, and did not see one in this article. In lots of math articles, you'll see a section "generalizations" and the examples there are usually all nonexamples of the original. So for instance in the opening there is a big chain of inclusions amongst types of integral domains with the idea that each inclusion is proper. This is also right after the example of the quaternions is given. I don't think more space than already given is warranted, but there is not a unique perfect choice of exactly which nonexamples are described. I think the current choice is pretty good. JackSchmidt (talk) 16:12, 12 October 2009 (UTC)

On the other hand, it may be of interest to mention that what is arguably the most important ring ever is not a field. I don't see this as a slippery slope. RobHar (talk) 16:19, 12 October 2009 (UTC)

Field extension

In the section 'Examples' under 'Rational and algebraic numbers' it says:

In the language of field extensions detailed below, Q[ζ₃] is a field extension of degree 3.

Isn't this a field extension of degree 2, since ζ satisfies the equation:

1 + ζ + ζ² = 0

where ζ and ζ² are the two primitive cube roots of unity? Namely, that everything written in the form:

a + b · ζ + c · ζ²

can in fact be written in the form

d + e · ζ

Apologies if I'm completely wrong here... Beast257 (talk) 15:41, 21 February 2010 (UTC)

You are completely correct. The article has been corrected. RobHar (talk) 18:10, 21 February 2010 (UTC)

Simple introduction

(Mentioned above,) there are soc and psych materials that resolve here, so I think that the first line should address that level of inquiry (what that would be, I don't know).--John Bessa (talk) 15:55, 18 May 2011 (UTC)

What exactly links here? Jakob.scholbach (talk) 21:36, 18 May 2011 (UTC)

What links here (more than 1000 links) --Nobar (talk) 17:28, 17 July 2013 (UTC)

Agree. The introductory portion can only be understood by people that already know what a Field is -- and even then, they might not understand it. Perhaps this is as intended, and an introduction for non-mathematicians should be maintained at Simple English Wikipedia.--Nobar (talk) 17:42, 17 July 2013 (UTC)

I very much agree that an elementary introduction which conveys the basic concept of a field to a reader at any level is absolutely necessary here. As it happens, there is already a section of the current article which accomplishes that goal reasonably well:

Intuitively, a field is a set F that is a commutative group with respect to two compatible operations, addition and multiplication, with "compatible" being formalized by distributivity, and the caveat that the additive identity (0) has no multiplicative inverse (one cannot divide by 0).

Dratman (talk) 21:06, 21 December 2013 (UTC)

"Connected"?

Recently this was added: "consisting of two connected abelian groups:". Is this 'connected' universal algebra lingo? It'd be nice if there were a wiki on 'connected group' in this sense, so as to distinguish between them and topologically connected groups. (I admit such an interpretation is unlikely, but no more unlikely than someone knowing what connected means in universal algebra.) Rschwieb (talk) 16:32, 24 May 2011 (UTC)

I believe the editor was using the term in the non-technical sense of "related". I think this addition is unnecessary and should be removed. RobHar (talk) 17:13, 24 May 2011 (UTC)

I don't quite follow you. Are you saying the reference to "connected Abelian groups" should not appear in the article? Dratman (talk) 21:13, 21 December 2013 (UTC)

I understand the objection to the word "connected". When I first read it, I assumed it was meant in the topological sense, as seen e.g. here. In fact it means that the two groups have almost the same set of elements. Maproom (talk) 07:47, 22 December 2013 (UTC)