Continuity of trace and norm

Question

Let $L|K$ be a finite extension of discrete valuation fields (not necessarily complete). Consider the classical maps:

$$\operatorname{Tr}_{L|K}:L\to K$$ $$N_{L|K}:L^\times\to K^\times$$

Are such functions continuous with respect to the valuation topologies?

Answer 1

Yes.

Let $N$ be a normal closure of $L$ over $K$, the valuation on $L$ can be extended to $N$ (not necessarily unique, fix one). It suffices to show that each $K$-embedding $\sigma: L \to N$ is continuous, we can extend $\sigma$ to $N$. Let $\mathfrak{P}$ be the prime ideal of $N$.

Note that $\sigma(\mathfrak{P})= \mathfrak{P}$. Since $\sigma:N\to N$ is additive, it suffices to show continuity at $0$, this is obvious: for $x\in N$, $$x \in \mathfrak{P}^{n} \iff\sigma(x)\in \mathfrak{P}^{n}$$