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?

  • $\begingroup$ What is an acceptable description of the splitting field? Note that you can find explicit roots for a cubic equation always.. $\endgroup$ – Asvin Mar 20 '16 at 23:02

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 $\mathbb Q(\lambda, \sqrt{-3\lambda^2-8})$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.