Talk:Irreducible polynomial/Archive 1

Archive 1

Methods for proving irreducibility

Does anyone else think that this page could use some examples of the different methods for reducing polynomials, or showing that polynomials are irreducible? The page on finding roots presumedly talks about such things, but none of the methods mentioned there are familiar to me. If I were to add some information on, say, how to find roots in polynomials and Eisenstein's_criterion, should they be added here or to the Root-finding_algorithm page? --Culix 03:27, 18 April 2007 (UTC)

I would say that an overview of methods of proving irreducibility belongs in this article, since root-finding_algorithm is about numerical methods for locating roots. Be careful, however, not to duplicate too many details that are already in the articles on specific methods - there is a list in the "See also" section, to which I have added rational root theorem. Gandalf61 09:40, 18 April 2007 (UTC)

Old discussion

Shouldn't $p_{3}(x)=x^{2}-4/3\,=(x-2/3)(x+2/3)$ be $p_{3}(x)=x^{2}-4/9\,=(x-2/3)(x+2/3)$? —Preceding unsigned comment added by 129.97.236.159 (talk • contribs) December 15, 2005

Yes it should be. It's of the form $(a+b)(a-b)=a^{2}-b^{2}\,$ —Preceding unsigned comment added by 83.194.35.3 (talk • contribs) December 20, 2005

In the Simple example section p_5 = x^2 + 1 is defined. Later p_3 = x^2 + 1 is mentioned. This polynom is the polynom p_5. Shouldn't it be renamed in p_5? —The preceding unsigned comment was added by 195.126.109.107 (talk • contribs) .

You're right, I've corrected it. Thanks for pointing that out. Next time you see something like that don't be afraid to be bold and correct it. toad (t) 16:46, 30 January 2006 (UTC)

Currently the article says "Over the ring $\mathbb {Z}$ of integers, the first two polynomials are reducible, but the last two are irreducible (the third does not have integer coefficients)". Shouldn't it say "the last three are irreducible"? I have changed the page; correct me if I'm wrong. --Culix 01:41, 18 April 2007 (UTC)

What?? Of course not. You shouldn't even be talking about the third one in the context of integer polynomials. Reverted. In fact, as written, the article doesn't even define irreducibility over integers, but then gives an example of it. ugh. — Preceding unsigned comment added by 140.254.93.114 (talk) 17:01, 30 April 2007 (UTC)

Algebraic Timeline

It seems to me that the timeline is incorrect. It says that people discovered ${\mathcal {A}}\cap \mathbb {R}$ then ${\mathcal {A}}\cap \mathbb {C}$ THEN calculus was invented then $\mathbb {R}$ and $\mathbb {C}$ were discovered. This doesn't seem accurate to me. Are you sure this timeline is correct? I've always been under the impression that calculus came before modern algebra (ie notions like algebraic closure etc...) I'm pretty sure that calculus was invented a good 70 years (at least) before algebra became mature enough for algebraic closures. —Preceding unsigned comment added by 74.192.193.127 (talk) 16:15, 9 February 2010 (UTC)

Counterexample

The article states that:

The irreducibility of a polynomial over the integers $\mathbb {Z}$ is related to that over the field $\mathbb {F} _{p}$ of $p$ elements (for a prime $p$). Namely, if a polynomial $p(x)$ over $\mathbb {Z}$ with leading coefficient $1$ is reducible over $\mathbb {Z}$ then it is reducible over $\mathbb {F} _{p}$ for any prime $p$. The converse, however, is not true.

I think it would benefit from an example of a polynomial that is reducible over Fp for all primes, but isn't for integers, if there are indeed any. —Preceding unsigned comment added by 87.99.27.160 (talk) 23:21, 11 April 2010 (UTC)

Monic or not?

Some authors I've seen seem to assume that all irreducible polynomials are necessarily monic, but this doesn't follow from the definition here (e.g. 2x^2 - 4 does not factor over the rationals as a product of two non-constant polynomials). Could we clarify this point? Thanks. Dcoetzee 12:16, 13 July 2010 (UTC)

Irreducibility over the integers

The definition that the article gives for reducibility over the integers is not universal. For example Stewart's Galois Theory has Definition 3.10 which says that the polynomial is reducible in a subring of C if it can be factored into two polynomials of lesser degree. Shouldn't both definitions be included? — Preceding unsigned comment added by 184.78.155.40 (talk) 19:59, 17 November 2013 (UTC)

Stewart's book is about Galois theory, and in Galois theory all polynomials are or may be supposed to be monic. In the case of monic polynomials, both definitions are equivalent. On the other hand, considering that irreducibility over the integers and irreducibility over the rationals are the same, as it results from Stewart's definition, would contradicts the general definition given in the section "Generalization". Moreover, the primitive part–content factorization is an important concept in polynomial factorization, because of the theorem "a primitive polynomial is irreducible over the rationals if and only if it is irreducible over the integers". How could it be stated with Stewart's restrictive definition? Note also that the modern approach of polynomial factorization, generated by the needs of computer algebra was certainly ignored by Stewart, when he wrote his book published in 1989.

Nevertheless, I would agree that a section "Irreducible polynomial over the integers or a unique factorization domain" is needed to avoid confusion. It should (and even must) contain a discussion to explain why old definitions like Steward's one are not convenient.

D.Lazard (talk) 14:05, 18 November 2013 (UTC)

What is a "non-trivial polynomial"?

The first sentence says that the factors must be "non-trivial polynomials", but "non-trivial polynomial" is not linked or defined. The article on Triviality (mathematics) does not mention polynomials, so it does not help. --50.53.53.71 (talk) 11:49, 21 September 2014 (UTC)

 Fixed D.Lazard (talk) 12:40, 21 September 2014 (UTC)

Thanks. --50.53.53.71 (talk) 17:41, 21 September 2014 (UTC)

Definition

The main definition now is "a polynomial is said to be irreducible if it cannot be factored into the product of two (or more) polynomials of positive degree, whose coefficients are of a specified type".

• First of all it is wrong: the constants are NOT irreducible polynomials.
• Then, it is ambiguous: what does "a specified type" means?
• The part "(or more)" is not needed — it complicates the definition.
• Finally, in all the books authors define irreducible polynomials by using the notion of field. Almost anybody who need the notion of irreducible polynomial knows what is field (at least main examples).
• Anyway we cannot leave the definition invented by a user. Or somebody can give a reference?

In the case of $\mathbb {Z}$ or unique factorization domain the definition of irreducible polynomial as an irreducible element is strange: by this definition the constant polynomial 2 is irreducible over $\mathbb {Z}$ o, while 2x is not. By definition of some authors it is not true. So, one needs to add a reference. NoKo (talk) 14:40, 12 October 2014 (UTC)

"Almost anybody who need the notion of irreducible polynomial knows what is field": Polynomial long division is commonly taught before fields and ring, and this suffices to introduce divisibility and irreducible polynomials. Thus your quoted assertion is wrong.

I agree with your other remarks, but, as this article does not satisfies the guidelines of WP: Manual of style, it is difficult to implement them in a way that improve the article. In particular, it is correct that the first sentence of the lead contains an informal definition, but, normally it should be clear that this definition is informal, and an accurate definition should appear in the body. Also, the lead is far too long. Having an informal lead and several accurate sections (one for each nature of coefficients) would allow to distinguish the cases, keeping the article accurate and easier to read for everybody. Also for polynomials over a ring, two definitions are used in the literature. Firstly, irreducible over the field of fractions, in the case of an integral domain, and secondly irreducible as an element, in the case of an unique factorization domain. Both definitions are equivalent for primitive polynomials over a unique factorization domain (this should appear in the article).

Are you willing to restructure the article in this way? D.Lazard (talk) 15:22, 12 October 2014 (UTC)

So, one need to rearrange the text in the beginning by adding the section "Definition" with formal definition first "over a field", then maybe over some class of rings. But even an informal definition should be correct. At least reader should see from the first line that the (invertible) constants and zero are NOT irreducible polynomials. This is a point, which many of them want to look in the article. If you are afraid of a contradiction with the definition "as irreducible element" maybe you can use some phrases like "non-trivial polynomial" and "specified type" to give the informal definition. This informalities are acceptable, I think, if the formal definition will appear in the section "Definition". To explain the situation in cases more general than the case "over a filed" (might be with references) is also important. NoKo (talk) 23:11, 12 October 2014 (UTC)

Is p3(x) really reducible over the integers?

In the text as it is now, it says that $p_{3}(x)=9x^{2}-3\,=3\left(3x^{2}-1\right)\,=3\left(x{\sqrt {3}}-1\right)\left(x{\sqrt {3}}+1\right)$ is one of the "first three polynomials" reducible over the integers and that "(the third one is reducible because the factor 3 is not invertible in the integers)".

It seems to me that p3(x) is not reducible over the integers because the factor sqrt{3} is not an integer.

Am I correct? Wandering-teacher (talk) 15:05, 22 March 2016 (UTC)

There is an irritating distinction being made in the article between irreducible polynomials over a field (like the rational numbers, where every coefficient has a multiplicative inverse) and irreducible polynomials over a ring (like the integers, where coefficients do not generally have multiplicative inverses). The claim made in the article is that in the latter sense, the polynomial is irreducible because it can be factored as 3 * (3x^2 - 1), and the factor 3 "counts" as a factor (because it does not have a reciprocal in the integers). The usual notion of irreducible polynomial (i.e., the one that I learned de facto in school) is irreducibility over either the rational numbers or the real numbers; both of these are fields, so you don't end up with silly distinctions like this.

The accessibility of the article would probably be improved by focusing more on the field case, at least in the first few sections. --JBL (talk) 15:24, 22 March 2016 (UTC)