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I've posted this question already on MSE and didn't get much out of it, so I hope it's OK to repost here. I'm an undergraduate trying to learn local class field theory from the corresponding chapter ...

Let $K$ be a number field and $\mathcal{O}_K$ be its ring of integers with unit rank $\geq 1$. For a given prime ideal $\mathfrak{p}$, shall we say that the map $\mathcal{O}_K^\times \to (\mathcal{O}...