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Let $p$ be an integer polynomial. Is $\frac{p(x) - p(y)}{x-y}$ always irreducible over the integers ?

mick
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  • What does this mean? E.g. for $p(x)\equiv 1$ or $p(x)=x$ you say $0$ or $1$ is irreducible over the integers!? – gammatester Oct 30 '15 at 12:28
  • Nonconstant polynomials – mick Oct 30 '15 at 12:28
  • $p(x)=x$ is non-constant! PS: I did not down-vote, I just want to know what you mean. E.g. do you see the expression as a polynomial in x and y? – gammatester Oct 30 '15 at 12:29
  • Have you tried anything? Even barring the obvious counterexample $p(x) = x$ there are others in very low degrees... Do you know the definition of irreducible at least? – Najib Idrissi Oct 30 '15 at 13:06

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Irreducibility is, in general, irreducibility over a field, but anyway, if we take $p(x)=x^4+x^2$ we have: $$ \frac{p(x)-p(y)}{x-y} = (x+y)\cdot(x^2+y^2+1).$$

Jack D'Aurizio
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