Degree of the Kummer extension - computing

Question

I have a question.

I want to calculate the degree of the Kummer extension.

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

[Page 3 in this publication]

What is the easiest way to do this?

score 1 · Accepted Answer · edited Apr 13 '17 at 12:20

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 $\frac{\phi(k)k}{2^s}$.

edited Apr 13 '17 at 12:20

Community

1

answered Mar 19 '17 at 20:38

Dietrich Burde

130,978

Why $2^s$ ? It depends if $g^{p/k} \in \mathbb{Q}(\zeta_k)$ for each $p | k$, right ? – reuns Mar 19 '17 at 21:11
Yes, the $s$ is as defined as in my other answer. – Dietrich Burde Mar 20 '17 at 09:55

Degree of the Kummer extension - computing

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