Previous theory: All the cyclic extensions of order $5$ are $\mathbb{Q}(\zeta_5)(\sqrt[5]{\gamma})/\mathbb{Q}(\zeta_5)$ where $\zeta_5$ is the generator of the group $\left(\mathbb{Z}/5\mathbb{Z}\right)^*$ and $\gamma \in \mathbb{Q}(\zeta_5).$ They are not Galois over $\mathbb{Q}$ in general, but using Elliptic Curves there is a way to find infinite Kummer extensions Galois over $\mathbb{Q}.$
It does not matter the way is used to find them, the important thing is that I found that for exemple the polynomial $x^{20} - 2x^{19} - 2x^{18} + 18x^{17} - 32x^{16} + 88x^{15} + 58x^{14} - 782x^{13} + 1538x^{12} + 1348x^{11}- 466x^{10 }- 894x^9 + 346x^8 - 114x^7 - 424x^6 - 88x^5 +214x^4 + 54x^3 + 14x^2 + 4x + 1$ defines a Kummer extension $E$ (it is cyclyc of order $5$, thus for the theory it is Kummer and is classificated).
So I want to find $\gamma$ such that $E=\mathbb{Q}(\zeta_5)(X^5-\gamma)$. Do you have any idea on how to do this?
EDIT: As user Johaness Huisman proposed me, this is the polynomial in a quoted form for being used on a mathematical software:
x^20 - 2*x^19 - 2*x^18 + 18*x^17 - 32*x^16 + 88*x^15 + 58*x^14 - 782*x^13 + 1538*x^12 + 1348*x^11 - 466*x^10 - 894*x^9 + 346*x^8 - 114*x^7 - 424*x^6 - 88*x^5 + 214*x^4 + 54*x^3 + 14*x^2 + 4*x + 1