Talk:Division ring/Archive 1

Semantic

Concerning the remark on the semantic effect in 'skew fields': in the theory of groups, a 'representation' is a special case of a projective representation.

--David.Gross 17:17, 24 September 2005 (UTC)

I've removed the "Semantics" section, because it appears to be patent nonsense in the second sense: "Content that, while apparently meaningful after a fashion, is so completely and irredeemably confused that no intelligent person can be expected to make any sense of it whatsoever."

First, I have no idea what the author means by "prefix" and "suffix," as any meaning used in linguistics fails to apply. Even if you wish to consider "skew field" as a single lexeme (a compound word), if "skew" is a prefix and "field" is a suffix, what's the root? The space character? It's pretty clear that "field" is a root, and "skew" is either a prefix or a lexically distinct modifier.

Even if you fix the terminology, the observation is so far from true that it's hard to imagine anyone could believe it for a second. Therefore, I'm giving the author Robinh the benefit of the doubt and assuming that this text was intended to say something completely different from what it actually says.

Reference-widening modifiers are common in all languages, including English, and particularly so within mathematics--see semigroup, nonassociative ring, Gaussian integer, and countless other examples.

Also, who exactly is the "Godemont" cited?

Here is the text that I removed:

===Semantics===

Skew fields have an interesting semantic feature: a prefix, here "skew", widens the scope of the suffix (here "field"). Thus a field is a particular type of skew field. This phenomenon appears to be rare in English, the only other example being Godemont's claim that tea is a particular kind of "leaf tea".

69.107.70.159 15:58, 6 March 2006 (UTC)

Hi.

The point I was trying to make was straightforward: a skew field is a particular kind of field. And I wanted to say that adding a qualifying word normally reduces the scope (eg "car" -> "red car") and here it is the opposite.

Sorry for any confusion

Robinh 16:26, 6 March 2006 (UTC)

Heh. You mean, of course, that a field is a particular kind of skew field. I think we could use considerably fewer words to make this point. -- Fropuff 18:58, 6 March 2006 (UTC)

Hi.

yes, I got it the wrong way round! Which underscores the point: such a construction is rare in English, alhough our anonymous friend 69.107.70.159 does point out a few examples in algebra. Thinking about this, perhaps the qualifying word, although extending the scope of the object, reduces the scope of the axioms (ie the field axioms are a particular kind of skew field axioms).

Robinh 08:26, 7 March 2006 (UTC)

I wanted to make the same point as "our anonymous friend". Actually, I had thought of the same examples (except for semigroup), and also manifold with boundary. You definitely shouldn't use "prefix", as there are a number of prefixes which widen the scope: a supercommutator is not a commutator, a q-factorial is not a factorial, a quasi-periodic function is not a periodic function (even in everyday English the prefixes quasi and semi widen the scope).

Regardless of how often it occurs, I don't find it very relevant. -- Jitse Niesen (talk) 10:21, 7 March 2006 (UTC)

This construction is less common than the opposite case, but not nearly so rare that it's worth pointing out in every case. Also, note that in a wide variety of constructions, neither case holds. A fake fur is not a fur, and a fur is not a fake fur either. Think of bubble tea, candy cigarettes, stone lions, cerebral lobes, near collisions, attempted coups, and expected losses. (Some of these are more subtle than others.) If you allow prefixes as well as adjectives (as your initial wording had it), things get even wilder.

There should be a place on wikipedia to discuss these semantic issues (if there isn't one already), but this page is not it. As Jitse Niesen points out, nobody who looks up "division rings" in an encyclopedia expects to read about adjectives.

Also, I'm still curious who Godemont is. (A Google search was no help. It's a bit odd to reference someone as an authority without providing any information as to who they are.)

Finally, I should point out that the term "reference-narrowing" is found far more often in computer science than linguistics. However, I thought it would be easier to understand than "subsective" for an interested layperson.

That tends to be typical in linguistics. If you want to read up on adjective semantics (which is actually a fascinating area at the moment), it's going to be tough going. There have been very few attempts to explain anything in linguistics to people outside the field. Without reading Richard Montague's "English as a Formal Language," a good introduction to the overall ideas and jargon of the generative school (Stephen Pinker's Words and Rules might do), and an introduction to modern semantics (I can't think of anything readable), almost everything out there is going to look like gibberish. (Here's a fun sentence from a relevant paper: "The 'X in [Det A X]' rule is often considered criterial, but perhaps can be reinterpreted as paradigmatic.")

Your anonymous friend (I have to find my username and password...) 69.107.95.34 12:02, 7 March 2006 (UTC)

Hello everyone.

Oh well, maybe it's not as interesting a fact as I thought. As for Godement, perhaps I should have said Roger Godement. When I get a minute I'll dig out my copy and quote him verbatim.

best wishes

Robinh 15:21, 7 March 2006 (UTC)

Roger Godemont

Ah. I thought you were citing a linguist. Roger Godemont is definitely a talented mathematician, but if he knew more about language he would have realized that supersective/antisubsective adjectives are just as common in English as in his own language. 69.107.87.12 06:26, 12 March 2006 (UTC)

Interlanguage links

I reverted DYLAN LENNON's edits because the interlanguage links ca:Cos (matemàtiques), da:Legeme (matematik), de:Körper (Algebra), eo:Korpo (algebro), es:Cuerpo (matemáticas), fr:Corps (mathématiques), it:Campo (matematica), nl:Lichaam (algebra), and pt:Corpos (matemática) all specify that multiplication is commutative, hence they are about field (mathematics) and not about division ring. The other links are in languages that I cannot easily read, or they give too little information. -- Jitse Niesen (talk) 08:19, 3 April 2006 (UTC)

Actually I think he's right on the French article, or at least partly right. Here's the relevant passage:

Un corps est une fr:structure algébrique consistant en un anneau unitaire dont tous les éléments non-nuls sont inversibles. Un corps est dit commutatif si c'est un anneau fr:commutatif, c'est-à-dire si sa multiplication est commutative. La plupart des corps que l'on manipule habituellement sont commutatifs, c'est pourquoi on ajoute souvent cette condition à la définition d'un corps.

La terminologie universitaire française, sous l'influence de l'anglais, fluctue ; souvent, elle considère que les corps sont tacitement commutatifs, les corps non commutatifs ou corps gauches étant déclarés tels, et parfois dénommés anneaux à (ou de) division.

For those who don't read French, the upshot is that the first paragraph, except for its last sentence, agrees with WAREL; the second paragraph equivocates on it a bit, saying that French usage has been influenced recently by English usage. --Trovatore 02:06, 5 April 2006 (UTC)

Yes, you're right. The Japanese article also seems to talk about both. -- Jitse Niesen (talk) 04:43, 5 April 2006 (UTC)

I've put back cs:Těleso (algebra), which is indeed about division rings. I checked pl:Ciało (matematyka), sk:Pole (algebra), and sl:Polje (matematika), all of them are explicitely about commutative fields. -- EJ 18:59, 5 April 2006 (UTC)

User:WAREL's latest sockpuppet changed the Japanese links, but also changed them in the ja:wiki, so I can no longer find the correct link for either Division ring or Field (mathematics). I'm thinking of deleting the ja links completely from both. Anyone here speak Japanese? — Arthur Rubin | (talk) 03:23, 10 October 2006 (UTC)

Indeed, something seems wrong. I think probably the English "skew field" means the same as the French fr:corps gauche, and the English Field (mathematics) means the same as the French fr:Corps (mathématiques). I hope someone can untangle this. – b_jonas 12:11, 4 February 2015 (UTC)

A bit of clarification

The sentence "If rings are viewed as categorical constructions, then this is equivalent to requiring that all nonzero morphisms are isomorphisms" could do with being a little more precise, I think. (Morphisms into the ring? Out of the ring? Endomorphisms?) Maybe whoever wrote it originally could explain what they had in mind and amend it. (Better than me trying to guess, and making a botch job of it.) -- Artie P.S. (talk) 10:22, 6 March 2008 (UTC)

A ring is a preadditive category (or Ab-category, if you prefer that) with one object. In this view, the morphisms of the category correspond to ring elements, composition of morphisms corresponds to ring multiplication, and addition of morphisms (which can be done because of the enrichment over Ab) corresponds to ring addition. So the translation of "every nonzero element has a multiplicative inverse" is "every nonzero morphism (of the category) has an inverse". Perhaps "In category-theoretic language, a division ring is a preadditive category with one object such that every nonzero morphism of the category is an isomorphism"? Or perhaps it doesn't need to be mentioned in the lede. Michael Slone (talk) 15:19, 6 March 2008 (UTC)

I've removed this sentence. It is correct, as Michael points out, but I think it's only going to confuse those who don't know category theory. Also, those that do understand the sentence are unlikely to understand division rings any better for it. It's just a restatement of the previous sentence. -- Fropuff (talk) 17:19, 6 March 2008 (UTC)

Definition of skew field

The first line says that a division ring (also called a skew field)_{not an exact quote}, however, IIRC, a skew field is a division ring that is not a field. In other words, a division ring is a field (exclusive or) a skew field. Albmont (talk) 18:10, 28 May 2009 (UTC)

• BTW, is there any example of a (non-commutative) skew field with characteristic zero that is not the quaternions over a subfield of the surreal numbers? Albmont (talk) 18:37, 28 May 2009 (UTC)

Conventions vary on whether division rings and skew fields can be commutative. French influenced sources will generally require skew fields to be non-commutative.

There are basically an enormous number of skew fields that are not particularly similar to quaternion fields. Look for "cyclic algebras" or more or less any abstract algebra text discussing central simple algebras, such as Jacobson's Basic Algebra. Probably the easiest example to understand is one of Dickson described in Lam's First Course in Noncommutative Rings: Take D to be the algebra over the rational numbers generated by x,y subject to xxx=2, xy=(yy-2)x, yyy=1+2y-yy with basis { 1, y, yy, x, yx, yyx, xx, yxx, yyxx }. The center is the rational numbers, and a maximal subfield is the dimension 3/index 2 subfield of the splitting field of x^7−1. By comparison, a generalized quaternion algebra has dimension 4 over its center and its maximal subfields are dimension 2 over the center. There are division algebras that cannot be associated to Galois groups like this, and there are even division algebras that are infinite dimensional over their centers. JackSchmidt (talk) 01:30, 31 May 2009 (UTC)

Thanks for the counterexample. I can see why there can't be a subskewfield of the quaternions that is isomorfic to this D - because if y were a quaternion, then {q1 + q2 y + q2 yy} would be a field, then y would be the solution of a cubic equation that falls into the casus irreducibilis, then y would be be real, then it would commute with x. I had this misinformation that every skew field was a subskewfield of the quaternions over something. Albmont (talk) 12:25, 2 June 2009 (UTC)

"A more complete comparison is found in the article Field (mathematics)"

This is written in the § about equivalents - of 'division ring' or 'field' - in other languages. But I don't find such a comparison there! (May-be it was there and has been removed or displaced?)

BTW I have created the interwiki link to French 'Corps (mathématiques)' after the reciprocal link has been corrected in the French article. --UKe-CH (talk) 13:47, 2 October 2009 (UTC)

the only non-trivial automorphism of complex numbers

Pertti Lounesto (2001, Clifford Algebras and Spinors, pp. 22–23) says that there are infinitely many field automorphisms for C, but only two R-algebra automorphisms. As this is very much not my area, I'm simply mentioning this for others, and hope it is not hopelessly misinterpreting something. —Quondum 22:06, 16 February 2014 (UTC)

Whoops… I ever forget about various pathological constructions based on Hamel bases and similar set-theoretical crap. Incnis Mrsi (talk) 06:33, 17 February 2014 (UTC)