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Consider $\mathbb Q_p^{\text{ur}}$ the maximal unramified extension of the p-adic numbers. Suppose that on $\mathbb Q_p$ we have the usual absolute value that extends $|\frac{a}{b}|_p=\frac{1}{p^{v_p(a)-v_p(b)}}$ to $\mathbb Q_p$. Now it is known that $|\cdot|_p$ extends uniquely to $\mathbb Q_p^{\text{ur}}$. Is $\mathbb Q_p^{\text{ur}}$ a complete space with respect to this absolute value?

I used notations and definitions as in here (pages 105-124).

My question is motivated by my attempt to find a a non-archimedean complete and discretely valued field, whose residue field is algebraically closed and I believe that $\mathbb Q_p^{\text{ur}}$ is a good candidate.

The other candidate that I tried was $\mathbb Q_p^{\text{al}}$, which has an algebraically closed residue field, but does not work for two reasons: First, the extension of $|\cdot|_p$ is not discrete w.r.t. $\mathbb Q_p^{\text{al}}$. Second, $\mathbb Q_p^{\text{al}}$ is not complete with respect to the extended absolute value.

Now, on the other hand, $\mathbb Q_p^{\text{ur}}$ has an algebraically closed residue field and as it is explained by Lubin here, it is also discretely valued. Now the only thing left to check is whether it is complete, but I am stuck here.

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No, it is not. $ \mathbf Q_p^{\textrm {unr}} $ has countable degree over $ \mathbf Q_p $, and it is a well-known fact that there are no Banach spaces of countable dimension over the ground field. (Hint: Baire category theorem.)

For your attempts, have you considered the field $ \mathbf C((T)) $?

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  • $\begingroup$ Thank you very much ! And in fact, I also believe the example that I tried can be fixed, if I replace the $\mathbb Q_p^{\text{ur}}$ by its completion instead. $\endgroup$ – user223794 Mar 29 '17 at 14:15
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It’s easy to see concretely that the maximal unramified extension is not complete:

Let $\zeta_1,\zeta_2,\zeta_3,\cdots$ be a sequence of integral elements in larger and larger unramified extensions of $\Bbb Q_p$. Like, for instance, $\zeta_m=$ primitive $(p^m-1)$-th root of unity. Then just write down the Cauchy series $Z=\sum_mp^m\zeta_m$. Obviously not in the maximal unramified exension, because that is an algebraic extension of $\Bbb Q_p$, and every element generates a finite extension. But $Z$ is not in any finite extension of $\Bbb Q_p$.

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