Let $K$ be a $p$-adic field, and $L$ an infinite abelian extension of $K$ containing $K^{ur}$. Let $\Phi: K^{\ast} \rightarrow Gal(L/K)$ be the local Artin map. Let $\pi$ be a uniformizer for $K$, and $K^{\pi}$ the fixed field of the subgroup of $Gal(L/K)$ generated by $\Phi(\pi)$. Is there an elementary way (say, without cohomology) of seeing that $L = K^{ur} K^{\pi}$?
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