When is $x^3+x+n$ reducible?

Question

I am trying to work out when $f=x^3+x+n$ is reducible over $\mathbb{Q}$, with $n\in \mathbb{Z}$. So far, I have reduced to irreducibility over $\mathbb{Z}$ by Gauss's lemma, and have that $f=(x+a)(x^2-ax+c)$. It is also clear that $f$ is reducible for $n=0,2,-2$. How should I proceed?

Answer 1

Let ${p\over q}$, $q>1$ a solution such that $gcd(p,q)=1$, we have ${p^3\over q^3}+{p\over q}+n=0$ implies that $p^3+q^2p+nq^3=0$ implies that $p^3=-q^2(p+nq)$ we deduce that $q$ divides $p$ contradiction.

We can suppose that $q=1$, we have $p^3+p+n=0$ implies that $p(p^2+1)=-n$, the polynomial is reducible if and only if $n=-p(p^2+1)$ where $p$ is an integer. Example, $p=2, n=-2(4+1)=-10$, we have $2^3+2-10=0$