Talk:Irreducible polynomial

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.

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)

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)