The following question is posed in the book "The USSR Olympiad Problem Book: Selected Problems and Theorems of Elementary Mathematics"
"Prove that if integers a_1, …, a_n are all distinct, then the polynomial
(x-a_1)^2(x-a_2)^2…(x-a_n)^2 + 1
cannot be written as the product of two other polynomials with integral coefficients"
I still haven't solved this in the elementary case, but I want to pose it in a more general setting. (I'll post the elementary answer in a bit, I want to try a bit more to figure it out)
EDIT: Now I have solved it, it's MUCH more trivial than I thought, major brain fart on my part
Suppose we have a ring R[x], and the polynomial written above is factorable in this ring. Suppose further that the coefficients are members of a subset of the field that the ring sits in, and that none of the a_i are equal (which means that some finite fields are of course out). Under what conditions IS the polynomial factorable into a product of two polynomials such that their coefficients sit inside the subset? Does the subset have to be algebraically closed?
(Is factorability even remotely easy in a noncommutative ring? I don't see a priori why a factorization would be unique either in a noncommutative ring).
Motivation: I'm doing research in mathematics education, and am interested in the metacognitive faculties of early college and high school pupils, which, for those not versed in metacognition, is the ability to separate oneself from "nitty-gritty" of the problem and think in more general terms. Alan Schoenfeld is the standard reference on this. I'm looking for problems that are good to pose to students to try and understand their thinking skills, and so am looking through particularly hard problems that do not require a strong background in mathematics. In this particular case I'd like to understand the problem in greater depth myself, and hopefully use my more general knowledge of the situation to aid in my study of how students think about such problems.
Hope this is interesting to someone, and that it isn't too specific.
Best Answer
I don't think this is a good problem for metacognition. Solving it is too contingent on what people have taught you about irreducibility.
Anyway, as for your general question, I am sure you can find a trivial example over Z/mZ for some composite m. Also, I can't tell whether you want a solution to the specific question, so here it is: suppose that the given polynomial is equal to f(x) g(x), where f, g have integer coefficients. Then f(ak) g(ak) = 1 for all k. Since f(x) g(x) has no real roots, neither does f or g, so they cannot take both the values +1 and -1. Therefore suppose WLOG that f(ak) = g(ak) = 1 for all k. On the other hand, at least one of f, g has degree at most n. The problem is straightforward from here.