Let $K$ be a characteristic $0$ field containing $\mu_n$ (the $n$-th roots of unity). Then it known that the map $K^{\times} / (K^{\times})^n \to \mathrm{Hom}(G, \mu_n)$ which sends $x$ to $\sigma \mapsto \frac{\sigma(x^{1/n})}{x^{1/n}}$ is well defined and is an isomorphism (here $G$ denotes the absolute Galois group of $K$).
Is it possible to give an explicit description of the inverse of this map ?
Choose $\chi\in\mathrm{Hom}(G,\mu_n)$. Let $H$ be the kernel of $G$ and let $L$ be the fixed field of $H$. For $\alpha\in L$, let $$ x(\alpha)=\sum_{\tau\in G/H} \chi^{-1}(\tau)\tau(\alpha). $$ Notice that $$ \sigma(x(\alpha))=\sum_{\tau\in G}\chi^{-1}(\tau)\sigma\tau(\alpha)=\sum_{\tau\in G}\chi(\sigma)\chi^{-1}(\tau)\tau(\alpha)=\chi(\sigma)x(\alpha). $$ So if $x(\alpha)$ is non-zero, you have constructed an element that transforms by $\chi$. Linear independence of characters tells you that there are lots of $\alpha$ that do this.