I have the following question.
Let $L/K$ be cyclic extension of local fields, $\sigma$ is a generator of $\text{Gal}(L/K)$, and $M/L$ is a finite abelian extension. Prove that:
$M/K$ is a Galois extension $\iff$ $\sigma(Nm_{M/L}M^*)= Nm_{M/L}M^*$,
$M/K$ is an abelian extension $\iff$ $\forall\alpha\in L^*$ $\sigma(\alpha)/\alpha \in Nm_{M/L}M^*$.
For the first question $\Longrightarrow$ is obvious. For the opposite implication it is sufficient to prove that if $M/K$ is not Galois then $Nm_{M/L}M^*\neq Nm_{\hat{M}/L}\hat{M}^*$, where $\hat{M}$ is some conjugate field to $M$ in $\overline{K}$, but I don’t know if this is true.
Any help would be appreciated