$E/F$ is an extension with radicals of order $n$

Question

We have that $E/F$ is an extension Kummer of degree $n$ and that $F$ contains a $n$-th unit root $\omega$ with $\text{ord} (\omega)=n$.

I want to show that $E/F$ is an extension with radicals of order $n$.

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I have found the following theorem:

Could we maybe use that theorem in this case? So, do we have to show that the Galois group of the extension has an exponent dividing $n$ ?