Let $\xi$ be the the fifth root of unity $\ne 1$, and let $z=\xi 2^{1/5}$ and $t=z^2+z^3$
I have to find an explicit expression for the minimal polynomial of $t$ over $\Bbb{Q}$.
I am stuck, I can prove that the splitting field of $X^5-2$ over $\Bbb{Q}$ is $\Bbb{Q}(\xi, 2^1/5)$ so the degree is $20$.
I can prove that $\Bbb{Q}(t)$ is included in $\Bbb{Q}(z)$. But then I am not sure what I have to "do".
You may simply examine the field $ \mathbf Q(z) \cong \mathbf Q(2^{1/5}) $. Using the obvious basis and writing out the multiplication map by $ 2^{2/5} + 2^{3/5} $ in matrix form gives
$$ \begin{bmatrix} 0 & 0 & 2 & 2 & 0 \\ 0 & 0 & 0 & 2 & 2 \\ 1 & 0 & 0 & 0 & 2 \\ 1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 \end{bmatrix} $$
Now, compute the characteristic polynomial of this matrix. Since $ 2^{2/5} + 2^{3/5} $ is irrational, it is a primitive element, thus the characteristic polynomial will be equal to its minimal polynomial.