# p-adic field extension of degree 2n without a subfield of degree 2?

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 $$[K:F] = 2$$.

• 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 $F(\sqrt{\varpi'}) \subseteq L$. – LSpice Aug 21 '19 at 2:10
• 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 $p = 2$. – LSpice Aug 21 '19 at 2:38
• 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. – Qirui Li Aug 21 '19 at 15:02
• 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. – LSpice Aug 21 '19 at 16:33
• 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. – 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.
• Thanks, but I need $[L:F]$ an even number. – Qirui Li Aug 21 '19 at 14:49
• @QiruiLi, yes, as I argued, you will have to take $p = 2$, in which case such an extension has even degree. – LSpice Aug 21 '19 at 16:32