[Math] Polynomial reducible modulo every integer

Question

Hi,

let $f\in\mathbb{Z}[X]$ be a monic polynomial. Assume that the reduction of $f$ modulo $m $ is reducible for all integers $m\geq 2$.

Q1: Is $f$ reducible in $\mathbb{Z}[X]$ ?

I've thought about this question without making substantial progress. If $p$ is a prime number , $p\nmid disc(f)$, then $f$ mod $p$ is separable, so if there is a nontrivial factorisation, we may find a factorisation into product of coprime monic polynomials. By Hensel's lemma, $f$ is reducible in $\mathbb{Z}_p[X]$.

If $p\mid disc(f)$, bad things could happen. For example, $X^4+1$ is reducible mod $2$, but irreducible mod $4$.

So I'm stuck here, and may be this approach will be not the right one but I have several related questions :

Q2: Assume that $p\mid disc(f)$. Does the fact that $f$ is reducible modulo $p^r$ for all $r\geq 1$ implies that $f$ is reducible in $\mathbb{Z}_p[X]$ ?

Q3: Let $f\in\mathbb{Z}[X]$ be a monic polynomial. Assume that $f$ is reducible in $\mathbb{Z}_p[X]$ for all prime integers $p$. Does $f$ is reducible in $\mathbb{Z}[X]$ ?

Thanks!

Greg

Answer 1

The polynomial $x^4-72x^2+4$ is irreducible over $\mathbb{Z}$, but reducible modulo every integer.

See E. Driver, P. A. Leonard and K. S. Williams Irreducible Quartic Polynomials with Factorizations modulo $p$ The American Mathematical Monthly Vol. 112, No. 10 (Dec., 2005), pp. 876-890