Currently, I am pondering about the following remark from Milne's Fields and Galois Theory:
Previously, Milne has shown that
$$ B(E)/F^{\times n} \to \operatorname{Hom}(\operatorname{Gal}(E/F),\mu_n), \quad: a \mapsto \left( \sigma \mapsto \frac{\sigma(a^{1/n})}{a^{1/n}} \right) $$ is an isomorphism if $E/F$ is a finite Galois extension of exponent $n$. So in this case, the pairing above is perfect. But I am not sure if this is also true if $E/F$ is not finite.
Question: Is this remark also true for infinite Galois extension of exponent $n$ (and if yes, why)?
Here is the section where Milne has shown it for the finite case if anyone is curious (note: $B(E) = E^{\times n} \cap F^\times$ here):
I am not sure here where we needed the finiteness of the extension in this explanation. I do not see immediately why it is needed here, so maybe this explanation also holds for the infinite case (but I may be wrong...).
The proof of Kummer's duality theorem certainly needs a finiteness assumption (look again at the details). For infinite abelian galois extensions $E/F$, you must adapt the hypotheses in an obvious way. To fix ideas, let us pick a prime $p$ and consider only infinite abelian pro-$p$-extensions (see example below). To extend Kummer's duality, we clearly need $F$ to contain the group $\mu_{p^{\infty}}$ of all $p$-primary roots of unity. As for the infinite pro-$p$ abelian extension $E/F$, consider it as an inductive limit of finite abelian $p$-extensions $E_n/F$. In characteristic $\neq p$, finite Kummer theory applies to show that $X_n:=Gal(E_n/F)\cong Hom(R_n,\mu_{p^n})$, where the Kummer radical $R_n$ is a finite subgroup of ${F^*/F^*}^{p^n}$. Taking appropriate limits, we get $X_{\infty}:=Gal(E/F)\cong \varprojlim Hom(R_n,\mu_{p^n})$. Writing $R_{\infty} $ for the inductive limit of the $R_n$, we see that $R_{\infty}$ is a subgroup of $F^* \otimes \mathbf Q_p/\mathbf Z_p$, hence a perfect compact-discrete duality $X_{\infty} \times R_{\infty} \to \mathbf Q_p/\mathbf Z_p$ (Pontryagyn duality).
I singled out this example because it's the starting point of the algebraic part of the so called Iwasawa theory, where $K$ is a given number field containing $\mu_p$, $F=K(\mu_{p^{\infty}})$, and $E/F$ has particular arithmetic properties (e.g. is unramified everywhere). Then $Gal(F/K)\cong \mathbf Z_p$ (pro-$p$-cyclic) acts continuously on $X_{\infty}$ via inner automorphisms. The extraordinary phenomenon is that this action gives rise to the construction of the $p$-adic zeta function (the $p$-adic analogue of the complex Dedekind function.)