Originally Posted by

**Opalg** Okay, here is the idea. Given a positive integer N, we want to find polynomials P(x), Q(x), with integer coefficients, each having height at least N, such that P(x)Q(x) has height 1.

Even for N = 2 this takes a bit of thought. Start with the polynomial

$\displaystyle (1-x^2)(1-x^4) = 1-x^2-x^4+x^6,$

which clearly has height 1. Now by repeatedly factorising the difference of two squares, write that polynomial as

$\displaystyle (1-x^2)(1-x^4) = (1-x^2)^2(1+x^2) = (1-x)^2(1+x)^2(1+x^2).$

Take $\displaystyle P(x) = (1-x)^2$ and $\displaystyle Q(x) = (1+x)^2(1+x^2).$ Then each of P(x), Q(x) has height at least 2, and their product has height 1.

Now do the same sort of thing for N = 3, looking at the polynomial $\displaystyle (1-x^2)(1-x^4)(1-x^8).$ Once you have sorted that one out, you will see how to do it for any N. (It's really a very simple-minded construction, nothing as sophisticated as cyclotomic polynomials!)