I need an example of a p-adic field extention $L/F$ of degree $[L:F]=2n$ without a subfield $K\subset L$ of degree $[K:F] = 2$.

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    $\begingroup$ Do you need this for some $n$, or one $F$ that has such an extension for every $n$, or …? If the residual degree of $L/F$ is even, then we may choose $K/F$ quadratic unramified; so assume it's odd. If $p \nmid 2n$, then the extension is tame, so one can write $L = E(\sqrt[e]\varpi)$ for some unramified $E/F$ and uniformiser $\varpi$ of $E$, where $e$ is the ramification degree of $L/F$. Since $E/F$ has odd degree, there is a uniformiser $\varpi'$ of $F$ such that $\varpi^{-1}\varpi'$ projects to a square in the residue field of $E$, and then $F(\sqrt{\varpi'}) \subseteq L$. $\endgroup$ – LSpice Aug 21 '19 at 2:10
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    $\begingroup$ In fact, if $p \ne 2$, then the maximal tame subextension of $L/F$ is also of degree $2n'$ for some $n'$, hence admits a quadratic subextension $K/F$ by the above argument; so this can only happen if $p = 2$. $\endgroup$ – LSpice Aug 21 '19 at 2:38
  • $\begingroup$ Hi LSpice, you are correct, the key here is the existence of maximal tame subextension. I understand now, you can leave your answer in the answer zone and I will accept it. $\endgroup$ – Qirui Li Aug 21 '19 at 15:02
  • $\begingroup$ Since my comments aren't an answer, just a restriction on when an answer can exist, and since @ChandanSinghDalawat did answer, I am reluctant to do that. $\endgroup$ – LSpice Aug 21 '19 at 16:33
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    $\begingroup$ Since it hasn't yet been mentioned, $x^4+2x+2$ is an explicit polynomial with Galois group $S_4$ over $\mathbf{Q}_2$, so $K = \mathbf{Q}_2[x]/(x^4+2x+2)$ is a degree $4$ extension with no quadratic subfields. $\endgroup$ – Puddles the turtle Aug 21 '19 at 23:46

For every prime $p$, every local field $F$ of residual characteristic $p$, and every integer $n>0$, there is a separable extension $L$ of $F$ of degree $[L:F]=p^n$ which does not have any intermediate extensions. All these $L$ can be explicitly parametrised. See for example Wildly Primitive Extensions.

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  • $\begingroup$ Thanks, but I need $[L:F]$ an even number. $\endgroup$ – Qirui Li Aug 21 '19 at 14:49
  • $\begingroup$ @QiruiLi, yes, as I argued, you will have to take $p = 2$, in which case such an extension has even degree. $\endgroup$ – LSpice Aug 21 '19 at 16:32

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