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I am interested to see what is currently known about the automorphisms of the topological field $\mathbb{C}_p$ of $p$-adic complex numbers (with respect to the $p$-adic topology induced by the $p$-adic norm extended from $\mathbb{Q}_p$). By automorphisms of topological fields I mean ones that are not only field isomorphisms but also are homeomorphisms.

Unless I've made some mistake it would appear that any automorphism of $\mathbb{C}_p$ is fixed on $\mathbb{Q}_p$ by continuity, so this would then boil down to finding the number of different ways to extend the identity map on $\mathbb{Q}_p$ to its algebraic closure (since it would then extend uniquely to $\mathbb{C}_p$ from $\mathbb{Q}_p^\text{alg}$).

I have not been able to find any literature on this, so maybe I am searching the wrong terms. I would very much appreciate being pointed in the direction of any relevant literature.

Many thanks

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    $\begingroup$ Through the observations you have made, you are asking about the absolute Galois group of $\mathbb Q_p$. This term should give you plenty enough search results. For instance for $p>2$ an explicit presentation of this group as a profinite group is known. $\endgroup$
    – Wojowu
    Commented Jun 12, 2021 at 21:41
  • $\begingroup$ @Wojowu If I remember correctly, the absolute Galois group would be the group of field automorphisms of $\mathbb{Q}_p^\text{alg}$ fixing $\mathbb{Q}_p$. Unless there is some guarantee that any such automorphism is also a homeomorphism, then presumably what I am looking for is a subgroup of the absolute Galois group of $\mathbb{Q}_p$. I'm looking at topological fields here rather than just the algebraic structure. $\endgroup$
    – Very Forgetful Functor
    Commented Jun 12, 2021 at 21:49
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    $\begingroup$ Yes, any such automorphism is automatically a homeomorphism. To see this, note that the automorphisms preserve $\mathbb Z_p$, and hence its integral closure $\overline{\mathbb Z_p}$. The topology is generated by elements $p^n\overline{\mathbb Z_p}$ for $n\in\mathbb N$ hence the result. $\endgroup$
    – Wojowu
    Commented Jun 12, 2021 at 21:53
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    $\begingroup$ See mathoverflow.net/questions/195907/automorphisms-of-mathbb-c-p $\endgroup$
    – KConrad
    Commented Jun 12, 2021 at 22:51
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    $\begingroup$ @AndréHenriques It is explained in detail in Neukirch, Schmidt, Wingberg Cohomology of Number Fields, section VII.5, specifically Theorem 7.5.14. $\endgroup$
    – Wojowu
    Commented Jun 12, 2021 at 23:53

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