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Finite extension of $K[[X]]$ and the norm

Let $R \colon= K[[X]]$ be a formal power series ring over a field $K$. We consider a monic polynomial $f(T) \in R[T]$ as follows$\colon$ $$ f(T) = T^e + c_{e-1}T^{e-1} + \ldots + c_1T + c_0. $$ ...
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