I currently reading an article by Tate in "Automorphic forms, representations and L-functions". The book consists of lectures given at Oregon state university in 1977. I am stuck at section (1.2) of the article wherein it is explained how the existence of Weil group of a field captures the cohomological class field theory. For the sake of completeness I will add some more details.
Let $W_F$ be the Weil group of $\overline{F}/F$ and the same notations for other extension of $F$ as well. Then just by using the axioms of the definition given in the article we get an exact sequence $$ 0 \rightarrow W_E/W^c_E \rightarrow W_F/W^c_E \rightarrow W_F/W_E \rightarrow 0 $$ where $W^c_E$ denotes the closure of the commutator of $W_E$ in $W_F(\supset W_E)$. Using the axioms of these groups, we can identify the above sequence with the short exact sequence $$ 0 \rightarrow C_E \rightarrow \star \rightarrow \mathrm{Gal}(E/F) \rightarrow 0. $$ Here $C_E$ is defined to be $\mathbb{A}^*_E/E^*$ if $E$ is a global field and $E^*$ if $E$ is a local field. Now, this short exact sequence above gives us an element say $\alpha_{E/F} \in \mathrm{H}^2(\mathrm{Gal}(E/F), C_E)$. Then it is claimed that for all $n \in \mathbb{Z}$, the map $$ \alpha_n(E/F) : \mathrm{H}^n(\mathrm{Gal}(E/F), \mathbb{Z}) \rightarrow \mathrm{H}^{n+2}(\mathrm{Gal}(E/F), C_E) $$ defined by taking cup product with $\alpha_{E/F}$ is an isomorphism.
I am unable to understand how this follows according to the arguments given in the article. As far as I can understand one needs the result that $\mathrm{H}^2(\mathrm{Gal}(E/F), C_E)$ is cyclic of order $[E:F]$(A reference for this result would be nice.). But that supposedly involves number theory and makes me wonder if that is the argument that Tate had in mind.