Prove that $\sqrt[3]{2} + \sqrt{2}$ is irrational

Question

Prove that $\sqrt[3]{2} + \sqrt{2}$ is irrational

This question is from Spivak’s book 18 of the third edition, but the solution is not intuitive. It suggests you to work out the first six powers of this number. Are there any more elegant approaches to this proof?

Answer 1

Let $y=2^{1/6}$ and $x=2^{1/3} + 2^{1/2}=y^2+y^3$. Suppose $x \in \mathbb Q$.

Since $y^3=x-y^2$, any power of $y$ can be expressed as a quadratic in $y$ over $\mathbb Q(x)=\mathbb Q$. In particular, from $y^6=2$ we obtain a quadratic equation for $y$ and so we have the contradiction that $[\mathbb Q(y):\mathbb Q]=2$.