Let $\alpha$ be a root of $x^3+x+1$ and $\beta$ be a root of $x^3+x+3$. Show that it is not possible that $\alpha\in \mathbb Q(\beta)$.
My proof :
Given $\beta$ is a root of $x^3+x+3$.
Thus $\beta^3+\beta+3 = 0$. Then $(\beta^3+\beta+2 ) + 1 = 0$.
If $\alpha\in\mathbb Q(\beta)$, then $\alpha = r_1 + r_2\beta$ for some rationals $r_1$ and $r_2$.
Also $(r_1+r_2\beta)^3+r_1+r_2\beta+1 = 0$.
Therefore for some rationals $r_1$ and $r_2$ we have $(r_1+r_2\beta)^3+r_1 + r_2\beta = \beta^3+\beta+2$.
But equating and solving such $r_1$ and $r_2$ doesn't exist.
Thus, $\alpha$ doesn't belong to $\mathbb Q(\beta)$.
Do you think my proof is right?
If not correct me or provide a better easier proof