Prove that $\sqrt[3]{2} + \sqrt[3]{4}$ is irrational.
My steps so far: I found that the polynomial $y^3-6y-6=0$ has roots $\sqrt[3]{2} + \sqrt[3]{4}.$ Can I use this to prove that $\sqrt[3]{2} + \sqrt[3]{4}$ is irrational? If so, how? I was thinking of using Proof by Contradiction, but I'm not so sure.