Let $k$ be a field of $char(k)=0$ und we consider an field exension $L/k$ with $[L:k]=n$.
Set $M:= L((t^{1/n}))$ and $F:= k((t))$.
I'm looking for a proof of following two statements:
1) If the residual field $k_F := \mathcal{O}_F/\pi_F $ equals $k$, then $M/F$ is totally ramified.
2) $char(k) =0$, therefore, $M/F$ is tamely ramified.
I'm not an expert on the field of algebraic number theory/ local field theory so unfortunately I have rather no idea how to show it. I was just curious about the both statements.
Nevertheless for 1) I tried follwing: $M/F$ is totally ramified iff $k_M= \mathcal{O}_M/\pi_M = \mathcal{O}_F/\pi_F= k_F$. We know $k_F=k$. Why $k_M=k$?
I know that if $p$ is the unique prime of $F$ then it ramifies via $p\mathcal{O}_M= q_1 q_2 ... q_d$ into primes in $\mathcal{O}_M$ but from here I'm stuck.
Could anybody give proofs for the two statements. The main goal is to show that $M/F$ is cyclic extension and I think that one can show this also directly but from didactical point of view (in order to learn some proof techniques from ANT) I would to prefer to go the way through steps 1) and 2).