Timeline for Automorphisms of the topological field $\mathbb{C}_p$ of $p$-adic complex numbers?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 12, 2021 at 23:53 | comment | added | Wojowu | @AndréHenriques It is explained in detail in Neukirch, Schmidt, Wingberg Cohomology of Number Fields, section VII.5, specifically Theorem 7.5.14. | |
| Jun 12, 2021 at 23:02 | comment | added | André Henriques | @Wojowu I would be interested to see the explicit presentation of this Galois group as a profinite group. Would you mind sharing? | |
| Jun 12, 2021 at 22:51 | comment | added | KConrad | See mathoverflow.net/questions/195907/automorphisms-of-mathbb-c-p | |
| Jun 12, 2021 at 21:53 | comment | added | Wojowu | 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. | |
| Jun 12, 2021 at 21:49 | comment | added | Very Forgetful Functor | @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. | |
| Jun 12, 2021 at 21:41 | comment | added | Wojowu | 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. | |
| Jun 12, 2021 at 21:40 | review | First posts | |||
| Jun 13, 2021 at 3:39 | |||||
| Jun 12, 2021 at 21:38 | history | asked | Very Forgetful Functor | CC BY-SA 4.0 |