We want to find the splitting field of $$f(x) :=x^3+3x^2+3x-4 $$ over $\Bbb{Q}$ and $\Bbb{Z}_3$.
Attempt. As usual, we are searching for all the roots in over $\Bbb{Q}$ and $\Bbb{Z}_3$.
In $\Bbb{Z}_3$, it is not difficult to observe that $f(x) = (x-1)^3$, so the splitting field is the same, $\Bbb{Z}_3$.
And I faced a difficulty in $\Bbb{Q}$. One can write our polynomial in the form $f(x) =(x+1)^3-(5^{1/3})^3$ and find all the roots with the obvious analysis. But, is there another more simple and elegant way?
Thank you.