Let $L/K$ be a finite extension of $p$-adic fields. Is the norm map $N_{L/K}: L \rightarrow K$ an open map?
At the very least, what is the easiest way (say, without using local class field theory) to see that the images of $L^{\ast}$ and $\mathcal O_L^{\ast}$ under the norm $N_{L/K}$ are open subgroups of $K^{\ast}$?
Let’s see whether I can give a sketch of an argument that will hold up.
To answer both of your questions, it should be enough to show that for $n$ big enough, $N^L_K(1+p^n\mathscr O_L)$ is open in $K^*$. Now for $n$ big enough, the $p$-adic logarithm maps $1+p^n\mathscr O_L$ onto $p^n\mathscr O_L$, isomorphically as Galois modules. The “big enough” should be $n\ge1$ for $p\ne2$, and $n\ge2$ for $p=2$. The map is one-to-one and onto, homomorphism, and respects Galois because its coefficients are in $\Bbb Q_p$.
In view of the above, the Norm on $1+p^n\mathscr O_L$ is isomorphic to the Trace on $p^n\mathscr O_L$, which I’m sure you can see is an open map.