Claim : Let $0<a<b<c<d$ be integers. Assume that $d$ is prime. Then, the polnomyial $$f(x)=ax^3+bx^2+cx+d$$ is irreducible over $\mathbb Q$
How can I prove this result without using $f(x)=(rx^2+sx+t)\cdot (ux+v)$ and comparing the coefficients, which requires several casea and is not quite easy ?
I am looking for a compact proof using finite fields or criterions like Eisenstein or similar approaches.
Since the degree is $3$, it is sufficient to prove that the polynomial has no rational root. So, maybe the rational root theorem can help, but I did not find a proof using this approach which is easier than comparing coefficients.
Does anyone have an idea ?