Talk:Polynomial ring

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?
2. Is the set S a ring of polynomials? Yes, trivially so, by definition.
3. 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.
4. 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 January 2005 (UTC)

The ring linas provides is not a polynomial ring (or ring of polynomials which is just another name for the same thing), by definition. The fact that such a ring appeared after shaking a bit certain ring of polynomials implies nothing about it being a polynomial ring. — Preceding unsigned comment added by 190.148.252.186 (talk) 02:48, 28 November 2010 (UTC)

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 January 2005 (UTC)

Quote from the article:

P(X)=X²+X=X(X+1)

Here the expression P(X) means function P taken of argument X, while the expression X(X+1) does not mean function X taken of argument X+1, but rather the product of X by (X+1). Why not use explicite multiplication signs to avoid this unclarity, here and elsewhere?

P(X) = X²+X = X·(X+1)

Bo Jacoby 20:53, 25 October 2007 (UTC)

I think there is something wrong with this:

${\displaystyle \mathbb {Z} [{\sqrt {2}}]=\{P({\sqrt {2}})\mid P(x)\in \mathbb {Z} [X]\}=\mathbb {Z} \cup ({\sqrt {2}}\mathbb {Z} )}$, where ${\displaystyle {\sqrt {2}}\mathbb {Z} =\{{\sqrt {2}}z\mid z\in \mathbb {Z} \}}$.

Shouldn't it be:

${\displaystyle \{P({\sqrt {2}})\mid P(x)\in \mathbb {Z} [X]\}=\{a+b{\sqrt {2}}\mid a,b\in \mathbb {Z} \}}$

and not the union of ${\displaystyle \mathbb {Z} }$ and ${\displaystyle {\sqrt {2}}\mathbb {Z} }$?

Alsosaid1987 (talk) 19:56, 14 June 2024 (UTC)

I have edited your post by replacing $...$ with <math>...</math>.

In fact there are two errors in the formula: the same "x" is sometimes uppercase, sometimes lowercase, and ${\displaystyle \cup }$ must be changed into ${\displaystyle +.}$ I'll fix this. D.Lazard (talk) 20:15, 14 June 2024 (UTC)

Thanks for the fixes! (I copied and pasted, but I should've remembered that the displays are weird when you're logged in, but not in edit mode.)

22:10, 14 June 2024 (UTC) Alsosaid1987 (talk) 22:10, 14 June 2024 (UTC)

I complained with edit . Then p_m ≠ 0 if m > 0 was removed, but it's probably not wrong, just confusing, should be included and explained? Or the p variables-list should end with some special variable instead of p_m, so it's less likely to confuse the reader. ~2026-26612-6 (talk) 15:42, 13 January 2026 (UTC)