Talk:Irreducible polynomial

Phrasing of coefficient constraint

Joel B. Lewis, maybe a discussion about this will be interesting. The wording "... is irreducible if the coefficients 1 and −2 are considered as integers" has, in my interpretation, two problems. The first is the obvious one: that it appears to try to draw a distinction between the integers as elements of the ring of integers, and the integers as elements of a subring of the reals. This is like claiming that the set of integers cannot be regarded as a subset of the reals (or, to belabour the point, that integers cannot be considered as being real numbers). The second is more subtle, and is my primary objection: the factorization constraint specifically applies to the coefficients of the factor polynomials (by definition, also to the polynomial to be factored, but that does not make a difference here). The wording as it stands states that constraint applies specifically and only to the specific coefficients of the polynomial to be factored, which would only be useful if the first problem did not exist. Do you at least see where I'm coming from? —Quondum 02:05, 7 October 2017 (UTC)

I'll change "if the coefficients 1 and −2 are considered as integers" into "if the polynomial is considered as a polynomial with integer coefficients". The formulation "if the polynomial is considered as a polynomial over the integers" is also fine for me, but may be considered as too technical or jargon. D.Lazard (talk) 08:21, 7 October 2017 (UTC)

I'm not sure the concern is properly addressed. One should not assume that the reader makes the "standard" assumptions that someone familiar with the topic does. The person familiar topic might read "considered as a polynomial with integer coefficients" to mean "all operations being in the ring of polynomials with integer coefficients", which is what is really meant. The person who makes no such implicit inferences might simply interpret it at face value: that the coefficients of the initial polynomial are integers. Face it, "considered as" is not a formally defined phrase in mathematics (and has even less meaning to the newbie); it is a cue to look for unstated inferences. What about "... is irreducible if it is to be factored as polynomials with integer coefficients"? —Quondum 12:25, 7 October 2017 (UTC)

In fact, the concern is not really about the sentence that has been discussed, but about the preceding one. I'll try to address this. D.Lazard (talk) 12:57, 7 October 2017 (UTC)

I have edited the lead in an attempt to clarify this. Feel free to improve my wording. This discussion is, in fact, about the distinction between irreducibility and absolute irreducibility, and it appears that absolutely irreducible was not linked here, I have fixed this. D.Lazard (talk) 13:42, 7 October 2017 (UTC)

My primary concern is essentially addressed by the clarification "the field or ring to which the coefficients of the polynomial and its possible factors are supposed to belong". I'm not convinced that absolute irreducibility is relevant to the discussion at all. As I understand it, all that is being said is that an irreducible polynomial has exactly one "nontrivial" (more technically, "prime") factor in the polynomial ring under consideration. The definition section is pretty clear on this, and has reasonably concise and understandable language. However, my primary concern of possibly incorrect initial interpretation has at least been addressed. —Quondum 14:10, 7 October 2017 (UTC)

Contradiction

Sentence 1 in the lead says

In mathematics, an irreducible polynomial is, roughly speaking, a non-constant polynomial that cannot be factored into the product of two non-constant polynomials.

The section "Simple examples" says

Over the integers, the first three polynomials are reducible (the third one is reducible because the factor 3 is not invertible in the integers)...

where the third one is

$p_{3}(x)=9x^{2}-3\,=3(3x^{2}-1)\,=3(x{\sqrt {3}}-1)(x{\sqrt {3}}+1)$

This seems contradictory, as the first quote implies that p₃ is irreducible over the integers. Loraof (talk) 17:29, 11 October 2017 (UTC)

This issue is related to the sections Irreducibility over the integers, What is a "non-trivial polynomial"?, Definition, and Is p3(x) really reducible over the integers? above. Ultimately the problem stems from trying to write an article that covers the cases of coefficients in a field and in other rings simultaneously. It does not seem like an easy problem to solve, but any attempt should probably begin by reading those talk-page sections. --JBL (talk) 21:40, 11 October 2017 (UTC)

If I understand correctly, $p_{3}(x)$ is irreducible according to one definition, but reducible according to another definition.

First definition in Irreducible polynomial#Definition:

A polynomial with integer coefficients, or, more generally, with coefficients in a unique factorization domain R, is sometimes said to be irreducible (or irreducible over R) if it is an irreducible element of the polynomial ring, that is, it is not invertible, not zero, and cannot be factored into the product of two non-invertible polynomials with coefficients in R.

$p_{3}(x)$ can be factored as $3(3x^{2}-1)$, and since $3$ is non-invertible, $p_{3}(x)$ is reducible over the integers according to this definition.

Second definition in Irreducible polynomial#Definition:

Another definition is frequently used, saying that a polynomial is irreducible over R if it is irreducible over the field of fractions of R (the field of rational numbers, if R is the integers).

$p_{3}(x)$ is irreducible over the rationals, so according to this definition it is also irreducible over the integers.

Is this analysis correct? If yes, we should probably refine the claims about irreducibility of $p_{3}(x)$: We should state that it is irreducible according to one definition, but reducible according to another, and (briefly) explain why. Chrisahn (talk) 13:21, 6 December 2018 (UTC)

Good point. I have edited the article for clarifying that the second definition is not used in this article. D.Lazard (talk) 18:04, 6 December 2018 (UTC)

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"Non-constant"

From Irreducible polynomial#Definition (regarding polynomials with integer coefficients):

Both definitions generalize the definition given for the case of coefficients in a field, because, in this case, the non-constant polynomials are exactly the polynomials that are non-invertible and non-zero.

If I understand correctly, this means that $p(x)=3$ would be considered a non-constant polynomial over the integers. I find it hard to reconcile this with any reasonable definition of constant / non-constant. Can we fix this? Chrisahn (talk) 13:34, 6 December 2018 (UTC)

This was ambiguous, and is (I guess) now clarified.

@D.Lazard: Thanks! I still find it strange that $p(x)=3$ would be considered a "non-constant" polynomial. What do you think? Chrisahn (talk) 18:25, 6 December 2018 (UTC)

$p(x)=3$ is never considered as non-constant, and I do not see any sentence of the article that could implies that it could be considered as non-constant. D.Lazard (talk) 18:39, 6 December 2018 (UTC)

@D.Lazard: But it's a non-invertible and non-zero polynomial over the integers, right? If that is correct, then it's also non-constant according to this sentence: "...in this case, the non-constant polynomials are exactly the polynomials that are non-invertible and non-zero." I agree that $p(x)=3$ should not be considered non-constant. I'd say that sentence is misleading. Chrisahn (talk) 22:51, 6 December 2018 (UTC)

The referent of "this case" in that sentence is "the case when the coefficients come from a field". That could be clearer. --JBL (talk) 00:35, 7 December 2018 (UTC)

Clarified. D.Lazard (talk) 02:48, 7 December 2018 (UTC)

Yeah, I thought "this case" referred to the case described in the previous sentence. Thanks for clearing this up! Chrisahn (talk) 16:49, 10 December 2018 (UTC)

Some awkwardness in the introduction

The introduction seems to make strong assumptions about the ring the polynomial is defined over. In particular, an irreducible polynomial is not necessarily the same thing as a prime element over a general integral domain, and the statement "a non-constant polynomial that cannot be factored into the product of two non-constant polynomials" is obviously only true for fields. While the definition section clarifies this, it seems unnecessarily misleading to the casual reader.

(Note: this is my first time using Wikipedia's talk pages.) Mathematician-at-heart (talk) 03:49, 4 October 2020 (UTC)

Your first post follows perfectly the recommendations of the manual of style.

I agree that "prime" and "irreducible" are not always equivalent. Moreover, "prime polynomial" is rarely used, except when proving the equivalence over a field. Thus, I have edited the parenthesis accordingly. On the other hand the statement "a non-constant polynomial that cannot be factored into the product of two non-constant polynomials" being a definition cannot be true or false. The precise definition, when the coefficients do not belong to a field, is too technical and not enough commonly used for belonging to the lead. This a reason for having "roughly" in the first sentence. The other reason for this "roughly" is the ambiguity of "cannot be factored", explained in the next sentences. D.Lazard (talk) 08:33, 4 October 2020 (UTC)

Again about the lede

I think it's worth talking about rewriting the lede - someone I was helping a little while ago was also thrown off by this and it looks like it's caused confusion at various times. I would prefer talking about the existence of a "non-trivial factorisation" and then clarifying what this means for integral domains and fields. Changes have been talked about above but I still think it's all unclear, at the very least it should be pointed out that $\mathbb {Z}$ is an ID not a field and so we are looking at the other criterion. Wanted to discuss here before making any major changes. --Caliburn · (Talk · Contribs · CentralAuth · Log) 19:45, 24 December 2021 (UTC)

I have added a paragraph in the lead for clarifying this. D.Lazard (talk) 17:27, 25 December 2021 (UTC)

"Irreducible over F_p implies irreducible over Z"

The section § Over the integers and finite fields claims the above. This is true for the second definition of irreducibility, where reducibles factor into two non-constant polynomials, but unless I'm mistaken about something, it seems false for the first definition. Counter example: 3x+3 is reducible over the integers because it equals 3(x+1), and 3 is not a unit. However, it is not reducible over F_5, because 3 is a unit there. FourierIsDeathier (talk) 09:33, 10 April 2022 (UTC)

 Fixed D.Lazard (talk) 10:12, 10 April 2022 (UTC)

The section Nature of a factor has ambiguous/incorrect language

It rambles around the idea that roots might have "no explicit algebraic expression". This is not what Abel-Ruffini Theorem says. Note that $\sqrt{2}$ is as algebraic as the root of every other polynomial. The section has no meaning unless it states correctly the type of expression that is claiming might or might not exist and which existence or not does not affect irreducibility. It seems that the intent of the section was to refer to expressions in radicals.

It is questionable whether the clarification offered by the section is needed. The definition(s) state where the coefficients are taken from and if the coefficients are the complex numbers, then the definition allows all complex numbers in the factorization, potentially even using non-algebraic numbers. Anyway, if the section is to exist, it should at least have the correct statement of what is claiming. 208.127.178.195 (talk) 19:08, 5 January 2023 (UTC)

Error in lead?

D.Lazard, could you check this statement in the lead for correctness?

"Most often, a polynomial over an integral domain R is said to be irreducible if it is not the product of two polynomials that have their coefficients in R, and are not unit in R."

The last constraint seems to exclude the factor polynomials that have even one unit coefficient, but this can't be right. I think the intention might have been "... and which are not unit in the polynomial ring", since this would have fitted the Definition section more closely. Note also that it is not really sensible to treat a constant polynomial an element of R as may have been intended, despite the natural embedding of R into the polynomial ring. —Quondum 21:25, 25 March 2023 (UTC)

As, over an integral domain, the total degree of a product of polynomials is the sum of the total degrees of the factors, the units of a polynomial ring are exactly the same as the units of the integral domain of coefficients. So, both formulations are equivalent. The one of the article is simpler, but if you find it confusing, be free of changing it. D.Lazard (talk) 21:46, 25 March 2023 (UTC)

I've tweaked the main thing that was bothering me. The secondary one (that you have responded to) may not be worth addressing as being natural enough. —Quondum 22:59, 25 March 2023 (UTC)

Reducible over every finite field?

The polynomial $X^{4}+1$ is given as a “simple” example of irreductible poly over $\mathbb {Z}$ and reducible over every finite field $\mathbb {F} _{b}$. Is that so?

The previous section gives the factorization $\left(x^{2}+{\sqrt {2}}x+1\right)\left(x^{2}-{\sqrt {2}}x+1\right)$, which is only possible if 2 is a square in $\mathbb {F} _{p}$ (a quadratic residue modulo p), that is p = 1 or 7 mod 8. Another possible factorization is $\left(x^{2}+{\sqrt {-2}}x-1\right)\left(x^{2}-{\sqrt {-2}}x-1\right)$, which is possible if -2 is a quadratic residue modulo p, that is p = 1 or 3 mod 8. The case p = 5 mod 8 is not resolved. Is $X^{4}+1$ polynomial really reducible in $\mathbb {F} _{5}$, $\mathbb {F} _{13}$, $\mathbb {F} _{29}$…?

self-reply Answer is YES. If p = 5 mod 8, (implying p = 1 mod 4) then -1 is a quadratic residue. Let's call r = √-1. $\left(X^{2}-r\right)\left(X^{2}+r\right)=X^{4}-r^{2}$. Et voilà. 2A01:E0A:D8B:7AF0:3E97:EFF:FEBC:B87B (talk) 18:51, 1 September 2023 (UTC)

There is no need to consider the congruences modulo p: if -1 or 2 is a quadratic residue, then the first or the third factorization holds. If none is a quadratic residue, then their product –2 is a quadratic residue, and the second factorization holds. D.Lazard (talk) 18:59, 1 September 2023 (UTC)