In Milne's notes on Class Field Theory, I.1.9, there is the following claim stated without proof:
Suppose $L/K$ is a finite abelian extension of local fields, and let $\phi_K:K^\times\to Gal(L/K)$ be the local artin map, which is surjective with kernel $Nm(L^\times)$. Then $L/K$ is tamely ramified iff $1+\mathfrak{m}_K\subset Nm(L^\times)$.
I can prove one direction: consider the composition $K^\times/Nm(L^\times)\to Gal(L/K)\to Gal(l/k)$, where $l,k$ are the residue fields; let $p$ be the residue characteristic. The first map is an isomorphism, and if $L/K$ is tamely ramified then the second map is surjective with kernel having order prime to $p$. Now if $\alpha\in 1+\mathfrak{m}_K$, then $\alpha^{p^n}\in 1+\mathfrak{m}_K^{n+1}$ (prove by induction on $n$), and so for large $n$ we have $\alpha^{p^n}=1$ in $K^\times/Nm(L^\times)$. On the other hand, $\alpha$ is a unit, so it maps to 1 in $Gal(l/k)$, and hence by hypothesis the order of $\alpha$ is prime to $p$. It follows that in fact $\alpha=1$, so $1+\mathfrak{m}_K\in Nm(L^\times)$.
However, I don't know how to prove the other direction (except when $k=\mathbb{F}_p$ and thus $u^{p-1}\in 1+\mathfrak{m}_K$ for any unit $u$ of $K$). Can someone suggest a proof?
Follow-up: is it true more generally that the conductor of $L/K$ (i.e. smallest $f$ for which $1+\mathfrak{m}_K^f$ is killed by $\phi_K$) is equal to the smallest $f$ for which the higher ramification group $G_f=0$ (where $G_f$ is the subgroup of $\sigma\in Gal(L/K)$ satisfying $|\sigma(\pi_L)-\pi_L|<|\pi_L|^f$)?