Find the splitting field without the explicit roots

Question

Let $f = X^3 + 2X -2$. I want to find the splitting field of $f$ over $\mathbb{Q}$. My problem is that the roots of $f$ are too complicated, see Wolfram Alpha. How can I find this splitting field?

Answer 1

The most "concrete" description, I can think of is this: Suppose $\lambda$ is a root of the polynomial $f(X)=X^3+2X-2$. You can infer the existence of this root from Kronecker's theorem, which constructs for every irreducible polynomial $f$ over a field $K$ an extension field $L\supset K$ in which $f$ has a root.

Then using long division, you can express $f(X)=(X-\lambda)(X^2+\lambda X+(\lambda^2+2))$. You can further factor the quadratic $(X^2+\lambda X+(\lambda^2+2))$ by using the quadratic formula $\frac{-\lambda\pm \sqrt{\lambda^2-4\cdot(\lambda^2+2)} }{2}=\frac{-\lambda\pm \sqrt{-3\lambda^2-8} }{2}\in \mathbb Q(\sqrt{-3\lambda^2-8}) $ for your favourite choice of square root in $\mathbb C$.

Then you can express the splitting field of $f$ as $\mathbb Q(\lambda, \sqrt{-3\lambda^2-8})$.