Let $K$ be a finite extension of the field of $p$-adic numbers which contains the $p$-roots of unity.
There is an isomorphism $K^{\times} / (K^{\times})^p \to \mathrm{Hom}(\mathrm{Gal} (\bar{K} / K), \mathbb{F}_p)$ which sends $x$ to the map $\sigma \mapsto \dfrac{\sigma(x^{1/p})}{x^{1/p}}$.
Call this isomorphism $\delta$.
Is there a way to decide whether $\delta(x)$ is a ramified or unramified character (depending maybe on the fact that $x$ is an integer in $K$ or not) ?
Let me call $\alpha$ the Kummer map $K^{\times} / (K^{\times})^p \to \mathrm{Hom}(G_K, \mu_p)$. (It's more canonical to use $\mu_p$ than $\mathbb F_p$, unless you have chosen a specific $p$th root of $1$ in $K$). Suppose that $\alpha(x)$ is unramified. This means that
$$\frac{\sigma(x^{1/p})}{x^{1/p}} = 1$$
for every $\sigma$ in the inertia group $I \subset G_K$. In other words, ${\sigma(x^{1/p})} = {x^{1/p}}$, or, what is the same, $I$ acts trivially on $K(x^{1/p})$. This means precisely that the extension $K(x^{1/p})/K$ is unramified. Since $v(x^{1/p}) = v(x)/p$, the extension will be unramified precisely when $v(x) \in p \cdot v(K^\times)$, in other words when the valuation of $x$ is divisible by $p$ in the valuation group of $K$. (Note that this is not the same as saying that $x$ is a $p$th power in $K$.)