Factor $X^5+6X^3+15X^2+3$ in $\mathbb{Q}\left (\sqrt{-3},\sqrt{-2}\right )\left [X\right ]$?

Question

I need to factor the polynomial $X^5+6X^3+15X^2+3$ in $\mathbb{Q}\left (\sqrt{-3},\sqrt{-2}\right )\left [X\right ]$, how may I do it?

Answer 1

Mathematica tells us that such a polynomial is irreducible over $\mathbb{Q}(\sqrt{-2},\sqrt{-3})$.
To prove it, we may take a prime $p$ for which both $-2$ and $-3$ are quadratic residues, like $p=43$.
If the original polynomial splits over $\mathbb{Q}(\sqrt{-2},\sqrt{-3})$, it has to split over $\mathbb{F}_p$. However, the original polynomial is irreducible over $\mathbb{F}_{43}$.

Answer 2

The polynomial $X^5+6X^3+15 X^2 + 3$ has Galois group $S_5$. It therefore can't be factored over any number field defined by a set of radicals (if it could, the factors would have degree $< 5$ and you'd get a solution in radicals).