I've seen some similar questions around but I'll post this up anyways. There's this beautiful polynomial: $$z^4-z^3-5z^2-z-6$$ and I am to factorise it into a irreducible polynomial in $\Bbb Q$, $\Bbb R$ and $\Bbb C$. But it looks super gross and I don't know where to start. I tried to factorise it with $z^2$ at the front but it it turned out to be a bunch of fractions and roots which I found suspicious.
I have this strong feeling that I might've forgotten some crucial theory/formulae to make my life much easier.
In $\mathbb{Q}$ and $\mathbb{R}$:
\begin{eqnarray*} z^4-z^3-5z^2-z-6 &=& (z^4-5z^2-6)-(z^3+z)\\ &=& (z^2-6)(z^2+1)-z(z^2+1) \\ &=& (z^2+1)(z^2-6-z)\\ &=& (z^2+1)(z-3)(z+2) \end{eqnarray*}
In $\mathbb{C}$
$$ ...=(z+i)(z-i)(z-3)(z+2)$$