Let $F$ be a field and let $\zeta$ be a primitive $n$-th root of unity in $F$.
Now I am trying to understand the following section from Milne's Fields and Galois Theory (page 73):
The part "Thus we obtain an isomorphism" seems to imply that the map $F^\times \cap E^{\times n}/F^{\times n} \to \operatorname{Hom}(G,\mu_n)$ is somehow derived from the previous cohomology sequence. However, this is the part where I am stuck. In particular, I do not see where $\operatorname{Hom}(G,\mu_n)$ comes from.
My first thought is that it has something to do with $H^1(G,\mu_n)$ but classes of crossed homomorphism are not quite the same as homomorphisms.
Could you please explain this part for me? Thank you!