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Where is the flaw in this argument with $p$-adic extensions?
I cannot find what I am missing in the following computation. Let $K=\mathbb{Q}_p(p^{1/{(p-1)p^{\infty}}})$ and $L=\mathbb{Q}_p(\zeta_{p^{\infty}})$, where $\zeta_{p^n}$ is a primitive $p^n$-th root ...
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