Talk:Polynomial ring

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Problem with this article: it confuses "a" ring R[X] with "the" ring R[X]. There are many polynomial rings, of which R[X] is just one example.

Dear Linas, please sign your comments. For that, you need to use for tildas, like this ~~~~.

Also, please put a period of the end of sentence. I mean your addition of "See also".

Now, about your question above. I think, all polynomial rings R[X] are isomorphic to one another. So you can think the ring is unique. Am I getting something wrong? Oleg Alexandrov 22:35, 8 Jan 2005 (UTC)

My objection is that "the polynomial ring" assumes that polynomial multiplication comes from ring multiplication. But there are other ways to define multiplication, in which case one gets other rings which are not isomorphic. For example the frobenius polynomial ring defines multiplication by composition; in this case, one gets a different kind of ring. It is NOT isomorphic because composition just works differently!

The correct thing to do for this article would be to fix it so that it defines what it means to multiply polynomials together. linas 06:44, 17 Jan 2005 (UTC)

Linas, I don't think you are correct. The Frobenius polynomials form a ring under addition and composition, for sure, but that makes them a "ring of polynomials", not a "polynomial ring". I think that "polynomial ring" means R[X] and its multi-variable analogs, and nothing else. --Zero 11:10, 17 Jan 2005 (UTC)

I agree with that. In the same way, there is only one field of real numbers. There are many weird ways of defining addition and multiplication for real numbers or for a subset of it, but those are not the field of real numbers anymore. Oleg Alexandrov 16:15, 17 Jan 2005 (UTC)

Well, then at minimum, there needs to be some sort of disambiguation. Books on number theory abound with rings of polynomials of all kinds of shapes and sizes, and none are isomorphic or homomorphic to that thing you are calling "the polynomial ring". It is a subtle slip of the English language to equate "the ring of frobenius polynomials" with "the frobenius polynomial ring". But "the frobenius polynomial ring" is "a polynomial ring" which is not isomorphic to "the polynomial ring". See the problem?

The definition of "the polynomial ring" needs to say that the multiplication is not just any old multiplication, but specifically "Cauchy multiplication" (ref article Cauchy product ??) and one needs to verify that the cauchy product is commutative, transitive over addition, etc.

Finally, if you want to use the word "the" instead of the word "a", there needs to be a proof of uniqueness. There must be some "XYZ's Theorem" that states that all such rings are the same thing. Its far from obvious. For example, it is "Abel's Theorem" that all abelian groups are isomorphic to modular arithmetic, but this theorem is not trivial. I assume the ring theorem is non-trivial as well, right? Or am I missing something? linas 16:17, 18 Jan 2005 (UTC)

Very trivial. First, a polynomial ring R[X] in one variable is by the abstract definition, the ring obtained from R by adding the symbol X so that it commutes with all elements in R, and that the quantities 1, X, X^2, X^3, ...are independent. Anybody can confirm this definition? In these conditions, given two polynomial rings R[X] and R[Y], there exists a unique R-algebra morphism between them mapping X to Y. This is quite simple to prove. Oleg Alexandrov 22:02, 18 Jan 2005 (UTC)

The operation that replaces every X by a Y is an isomorphism from R[X] to R[Y]. That is trivial to prove and too obvious to mention. No other uniqueness result is needed. --Zero 09:14, 19 Jan 2005 (UTC)

Contents 1 Trivial proofs? 2 Frobenius Ring 3 Philosphy of wikipedia 4 Definition of polynomials

[edit] Trivial proofs?

OK, It may be trivial to you but it it is anything but to me. Lets try to work out a specific example. Lets consider the ring of all polynomials with integer coefficients: I think you'd call it Z[X]. Now consider the set S = {Z[X] mod (x+1)}.

1. Is the set S a ring? I think so. Its additive unit is 0, its have additive inverses, and is closed under addition. It has a multiplicative unit (x+1): every polynomial P(x) times (x+1) equals P(x) mod (x+1). Its closed under multiplication. That makes S a ring. Or am I missing something?

1. Is the set S a ring of polynomials? Yes, trivially so, by definition.

1. Accodring to the "Zero-Oleg trivial theorem", there exists some ring R such that S is isomorphic to R[X]. What is that ring R? I dunno, and it is not trivially obvious to me at this time.

1. How many isomorphisms exist between S and R[X] ? Again, not obvious to me. One? More than one? Some formula involving Euler's totient? What is that formula?

linas 19:45, 19 Jan 2005 (UTC)

You have a point. But the problem is, you miss the point of your discussion. You say you hack computers. Let me put it in that language.

There is only one default addition for the int type in C++. Only one. Now, the addition operator can be overloaded, redefined, whatever. But, the default operation is only one.

Now, the polynomial ring, the default polynomial ring, is the one where the addition and multiplication is defined the way you learned in high school, or rearlier. Now, you can redefine multiplication, but that is not the polynomial ring anymore, OK? Please understand this. That's another ring, still made up of polynomials, but not the polynomial ring. Oleg Alexandrov 20:50, 19 Jan 2005 (UTC)

Yes, but ... in the above example, the concept of multiplication is not redefined, that's why I picked this example. It is exactly the same multiplication that Z[X] uses. Is it "the" polynomial ring for some value of R?

I know from group theory that it can be very hard to figure out if two groups are the same group or not, if they are isomorphic to each other. This was the whole point behind abelian groups: you've got this big overarching theorem that says abelian groups are always just modular arithmetic, and if you have an abelian group, no matter how complex it may seem to be, you will eventually succeed in identifying which abelian group it is.

Here, you and zero seem to be implying that its somehow "easy" to figure out when a polynomial ring is "the" polynomial ring R[X] for some R. Is there some theorem I can use to find out if/when "a" polynomial ring is isomorphic to "the" polynomial ring for some R?

Lets assume I never try to redefine multiplication, say that I stick to standard cauchy product multiplication. That is, I want to only consider those polynomial rings that are generated by modulo arithmetic. Is there ever a situation where R[X] cannot be found, or is it always possible to identify R? Is there an algorithm for computing R? What is that algo? linas 23:31, 19 Jan 2005 (UTC)

Your S is not the polynomial ring R[X] for any ring R. This because in a ring R[X] one never has 1, X, X^2, etc be linearly dependent. Your S is the quotient of the olynomial ring Z[X] by the ideal which makes that identification modulo X+1 possible. Oleg Alexandrov 23:40, 19 Jan 2005 (UTC)

OK, well, I didn't know that the x^n have to be linearly independent. That wasn't an obvious assumption, but OK, my mistake. Now that I think about it, I guess that would make S a module (mathematics). So that makes sense. But still, common use is not to say "consider the module blah blah blah"; its always "consider the polynomial blah blah blah". Certainly, the engineering specs call them polynomials, and not modules. I'll bet that even most math books call them "polynomials" even when they really mean "modules"; it just seems to be a fairly common usage.linas 05:17, 20 Jan 2005 (UTC)

Dang, you've got me running in circles now. I re-read Ideal (ring theory), as suggested, and it seems to be saying that the quotient of a ring mod ideal is a ring, not just a module. So that means that the quotient of R[X] modulo p(x) **is** a ring, not just a module. So that means that the set S is a ring ... of polynomials ... its just not "the" ring of polynomials. So the original example stands: S is "a" ring of polynomials that is *not* isomorphic to "the" ring of polynomials and so zero's "trivial" theorem is in fact not a theorem at all, its just plain false! There is no such theorem!

I really dislike being disabused by you guys, pulling out shit like "theorems that are too trivial to prove", when in fact they are bullshit, and just plain old incorrect! linas 05:17, 20 Jan 2005 (UTC)

[edit] Frobenius Ring

Oleg, I think you are totally unrealistic. There is nothing elementary about polynomials, it is a big complex topic. The article just barely scratches the surface. And as to calling the frobenius polynomial ring exotic, it is defined on PAGE 3 of David Goss's 400-hundred page book "Basic structures of function field arithmetic", ... page 3 ... there is nothing exotic about it, its polynomials 101. linas 16:19, 18 Jan 2005 (UTC)

Well, I know who David Goss is, I know what he works on, and why. And I didn't know this topic. So I beg to differ. Charles Matthews 17:16, 18 Jan 2005 (UTC)

Well, I *don't* know him. Never heard of David Goss until I bought his book, and I still haven't heard of him in any other context. I was merely trying to solve some unrelated problems, and it seemed that this particular book might be vaguely related to the topic I was working on. On pages 1 & 2 he defines these polynomials, and on page three he says "some authors call this the ring of frobenius polynomials", and the rest of the book seems devoted to their study. I assumed that things on pages 1,2,3 of a fat book were not considered to be "esoteric". Maybe you are thinking of a different David Goss? linas 19:45, 19 Jan 2005 (UTC)

What the isomorphism between R[X] and the frobenius ring is, I have no clue. It is not trivial, to me, a novice. Goss doesn't seem to mention any isomorphism; maybe its so trivial that its assumed that everyone knows it, but .. well, I don't get that from reading the first chapter. linas 19:45, 19 Jan 2005 (UTC)

[edit] Philosphy of wikipedia

Dear Linas, please do not think we are just a bunch of people here who want to dumb down math. The aim of writing Wikipedia articles on math is not to have an encyclopedia of all things known to mankind. The purpose is to have very gentle, very elementary articles giving some insights in what math is about to the 99% of people who are not specialists. Now, there is room in Wikipedia for complicated articles too. But there should be a clear separation of simpler articles and more complicated articles. Just because I know certain things about say "Optimization", does not mean I need to go to that page and bring it "up to date" as far as the science or art of it is concerened. Oleg Alexandrov 18:16, 18 Jan 2005 (UTC)

Well, I'm one of the 99% who are not specialists. I am not a mathematician. My day job is to hack computers. This is a recreational, spare-time hobby activity for me. A year ago, I didn't know number theory from a hole in the ground. So as I follow my nose, I have to hope that what I find on mathworld and on wikipedia is reasonably accurate and complete.

When I read, I take notes: that this is an X and that is a Y, basic definitions and stuff, nothing complex. Frankly, I am not that smart. Till now, my notes have always been on scraps of paper. But suddenly there's this possibility that my notes could be in wikipedia, and so when things aren't clear, or they're vague, or the thing I'm taking notes on is clearly incomplete, then someone else will come along and fill in the empty spots in my notes. That would really help me.

Nothing I've added to wikipedia seems to be "cutting egde"; from what I can tell, everything I'm working on now seems to have been figured out in the 19th century. Its all named after people who died 100 years ago ... I don't know how to be more "dumbed down" and "aimed at the generalist" than that. Polynomial rings seems to be one of the 100-200 year old topics that are wildly relevent today. Each global positioning system (GPS) satellite broadcasts a unique polynomial; its at the core of how they work. Polynomial rings are used for error correcting codes in computers and in cell phones, etc. So its definitely neat stuff, but ... hey, its *not* cutting edge. I had absolutely no effing clue that all these polynomials were uniquely isomorphic to some R[X] for some ring R, and I don't think any other novice reading this article will know that either. (The GPS spec makes no mention of this, nor do the radio communications books I've read). Omitting this kind of proof, no matter how trivial it may seem, is a dis-service.

Dear Linas. I spent uncountable times talking to you in the last several weeks. Let me put it in short language what I think the heart of the problem is. It is not so much whether it is complicated stuff or not. The problem with your contributions are

• You do not think long enough before you insert your contributions.

• You do not read carefully the article as a whole after you insert your contributions.

• You do not have a good taste about what a good Wikipedia article is.

In the Polynomial ring article you made a change (removed the commutativity assumption), but you did not read the paragraph right below, which dealt with the uncommutative case. Then you would have realized your addition was not appropriate.

In the modular arithmetic article you made childish mistakes, and after I fixed one, you reverted it.

There are other examples, like my earlier deletions of your stuff, when you restored them and told me to not mess up with what you write because I don't know what I am doing. Eventually you deleted that stuff yourself.

In short, at least in several cases your contributions actually degraded articles. Please take a while to think of the three items above. Oleg Alexandrov 21:06, 19 Jan 2005 (UTC)

OK, so I make childish mistakes. Most recently, I accidentally confused a polynomial ring with a module. Sorry. They are alike in so many ways. But at this point, I wouldn't know how to edit the article on polynomial rings to say "hey duude, that thing you think is a polynomial ring? Well, duude, its probably not, its a module".

And similarly, the article module (mathematics) doesn't ever say that (for example) "the ring R[X] modulo a polynomial p(x) is a module". I'm not sure, but isn't *every* R[X] modulo p(x) a module? How should I ask the authors of the article on polynomial rings to state this? How do I ask the author of the article on modules to add this info? linas 01:15, 20 Jan 2005 (UTC)

It would not make a lot of sence to write articles guiding against all possible mis-conceptions (I know you had only one, but others can have other ones). I would suggest that you read the article Ideal (ring theory), and especially, the part about quotient ring in there. You are very right, a hell of a lot of modules are just quotients by some ideal. Oleg Alexandrov 01:54, 20 Jan 2005 (UTC)

You have a point about the polynomial ring article not being very complete. For example, something must be said about the nature of a polynomial in abstract algebra (as opposed to real analysis, see polynomial). I will get to this sometime soon.

But you should also not make the mistake of considering Wikipedia a serious reference about math (or trying to make it one). The encyclopedia format has its limitations (and they are good limitations). Oleg Alexandrov 04:07, 20 Jan 2005 (UTC)

Why not? Mathworld takes itself as a serious reference, and for a while was widely celebrated, until the great debacle. Should I be applying for a mathworld editorship instead?

The only other possible reference, the follow-on to Abramowitz and Stegun, is a decade late, and presumably will never be finished. I see no reason why Wikipedia couldn't/shouldn't be a replacement for Abramowitz & Stegun. Although, clearly, the idea of "anybody can edit anything" is trouble when accuracy is important... linas 05:25, 20 Jan 2005 (UTC)

See Wikipedia:WikiProject Mathematics and its talk page, where you can ask the hard questions. Just think carefully of what you ask :) Oleg Alexandrov | talk 10:38, 20 Jan 2005 (UTC)

[edit] Definition of polynomials

Coming back to this after two days away, I think that the definition still needs work. The problem is that people will think of polynomials as representing functions: such as X^2+X representing the function f from R to R defined by f(x)=x^2+x. However, thinking of R[X] as a set of functions from R to R is a mistake for several reasons. The main one is that several different-looking polynomials can be the same function, for example X^2+X and 0 are always equal if R is GF(2), but they are different elements of R[X]. We can also note that in the case of finite R every function from R to R can be written as a polynomial. I think the right way to think of R[X] is the one given here: http://planetmath.org/encyclopedia/PolynomialRing.html . That is, the polynomials are just convenient ways to write down sequences. That definition also makes it clear that R[X] and R[Y] are not just isomorphic but identical. --Zero 12:06, 21 Jan 2005 (UTC)

You are certainly right. I alluded to this problem above. I will soon get to explaining what exactly a polynomial is. Oleg Alexandrov | talk 16:26, 21 Jan 2005 (UTC)

I was ready to work more on this article, but when I checked out the polynomial page, I found an excellent section there about polynomials in abstract algebra, see Polynomial#Abstract algebra. So what should we do? Shorten the thing over there and move most stuff here? Do nothing? Make a copy of that stuff? The only thing missing on that page is the formal construction of the polynomial ring, by means of sequences of finte length, as Zero says above. but I am not even sure that is necessary. Help! Oleg Alexandrov | talk 01:25, 23 Jan 2005 (UTC)

I hadn't noticed that. Actually there is another thing missing from Polynomial#Abstract algebra: the multivariate case. I think the simplest formal way to define R[X,Y] is that it means (R[X])[Y]; whether that is the easiest definition to understand, I'm not sure. As to what to do with this article, I think we should fix our own definitions to be consistent, then add a "for more information see" link from polynomial to here. I don't think we should reduce the material at polynomial. --Zero 02:27, 23 Jan 2005 (UTC)

Well, fixing the defintions to be consistent probably means copying that thing over, and then work from there. Other ideas? By the way, what is written in Polynomial#Abstract algebra is indeed neat, I could not have written it so well myself. Oleg Alexandrov | talk 02:30, 23 Jan 2005 (UTC)

Go for it. --Zero 06:08, 23 Jan 2005 (UTC)