Let

$\displaystyle f(x) = x^4+ax^3+bx^2+cx+d$

be a polynomial in $\displaystyle \mathbb Q[x] $ which has no rational root.

Show that $\displaystyle f(x) $ is irreducible in $\displaystyle \mathbb Q [x] $

if the cubic resolvent

$\displaystyle g(y)=y^3-by^2+(ac-4d)y-a^2d+4bd-c^2 $ has no rational root.