Degree of the Kummer extension - computing

Question

I have a question.

I want to calculate the degree of the Kummer extension.

$D_{g}(k) = (\mathbb Q(g^{1/k}, e^{2\pi i /k}) : \mathbb Q)$

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What is the easiest way to do this?

Answer 1

This is the degree of the splitting field of the polynomial $X^k-g$. The degree has been computed in general, see the answer here, namely $\frac{\phi(k)k}{2^s}$.