Suppose $K/\mathbb Q_p$ is a finite extension, with ramification index $e$ and inertia index $f$. I want to ask whether there exists a subextension $L/\mathbb Q_p$ totally ramified of degree $e$?
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$\begingroup$ Now it is a duplicate of this question. There this field is shown as an example of a degree 6 extension of $\mathbb{Q}_2$ with $e=2$ but only a unique quadratic subfield, which is unramified over $\mathbb{Q}_2$. $\endgroup$– Chris WuthrichCommented Jun 29 at 9:47
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$\begingroup$ @ChrisWuthrich Thank you very much. $\endgroup$– RichardCommented Jun 29 at 9:55
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$\begingroup$ @ChrisWuthrich It is not a duplicate, that question is asking about a weaker property. This question is much more basic. The field $K = \mathbf{Q}_5(\sqrt[4]{50})$ is cyclic of degree $4$ over $\mathbf{Q}_5$ (the splitting field of $x^4 - 50$ and $\sqrt{-1} \in \mathbf{Q}_5$) with $(e,f)=(2,2)$ but it has a unique quadratic subfield $\mathbf{Q}_5(\sqrt{50}) = \mathbf{Q}_5(\sqrt{2})$ which is unramified (as it has to be), so no such $L$ exists. $\endgroup$– user491858Commented Jun 29 at 17:56
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