Let $\xi$ be the the fifth root of unity $\ne 1$. Let $z=\xi2^{1/5}$ and $t=z^2+z^3$.
I would to prove that $\Bbb{Q}(t)=\Bbb{Q}(z)$. As $\Bbb{Q}(z)$ is the smallest field containing $z$ it contains $z^2$ and $z^3$ and so $z^3+z^2$.
However, the converse seems more 'difficult', is it possible to prove it without calculus?
The only way I see is to compute the first powers of t: $$t^2=z^4+2z^5+z^6=4+2z+z^4$$ and $$t^3=z^6+3z^7+3z^8+z^9=z+6z^2+6z^3+z^4=z+z^4+6t.$$ Now it is clear that $$t^2-t^3+6t-4=z,$$ hence the converse inclusion follows immediately. But if you meant exactly this by calculus, then I don't know any direct result (and actually don't expect any, as extensions of this type can vary a lot by a simple change of constants).