Let $k$ be a field. We consider the field extension $k(t^{1/n})/k(t)$. My question is why and how to see that it is cyclic, therefore $Gal(k(t^{1/n})/k(t))$ is a cyclic group.
There is one case that is easy:
If we assume that $k(t)$ contains a primitive $n$-th root $\zeta_n$ and futhermore $k(t^{1/n})= k(t)[X]/(X^n-t)$
Kummer thery tells that the extension has indeed cyclic Galois group.
By how to argue if $k(t)$ doesn't contain a primitive $n$-th root? Here Kummer can't help. Any ideas?
Futhermore, what about if we play the same game with the extension $$k((t^{1/n}))/k((t))$$ therefore fractions of $k[[t]]$.
I suppose that this extension has the same Galois group as the extension $k(t^{1/n})/k(t)$ above. Does anybody know how to see it formally/ maybe a theorem providing this result?