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I'm a graduate student studying now for the first time class field theory. It seems that how to teach class field theory is a problem over which many have already written on MathOverflow. For example ...

Let $K$ be a $p$-adic field with algebraic closure $\overline{K}$. Then if $K^\text{nr}$ is the maximal unramified extension of $K$ in $\overline{K}$, there is an explicit invariant map: $$ H^2(\...