Timeline for Place stabilizers for the absolute Galois Group
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Dec 2, 2013 at 4:34 | comment | added | Adam Hughes | Thanks Keenan, I appreciate it! I had actually already worked out the generalization for when you can apply Krasner's lemma, but it's good to see I was not wrong in my guess that it generalized. | |
| Dec 2, 2013 at 4:06 | comment | added | Keenan Kidwell | Dear @Adam, I edited in an answer to your questions. | |
| Dec 2, 2013 at 4:05 | history | edited | Keenan Kidwell | CC BY-SA 3.0 |
Answered question in comments
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| Dec 2, 2013 at 1:19 | comment | added | Adam Hughes | That $\varphi$ is continuous is easy, I suppose, from the inverse limit definition of the groups. Am I correct in thinking $\varphi$ is well-defined by the FTGT, or am I missing the mark on something? | |
| Oct 30, 2012 at 11:09 | comment | added | Keenan Kidwell | I guess this procedure works equally well for Archimedean primes. | |
| Oct 30, 2012 at 11:08 | comment | added | Keenan Kidwell | an absolute value whose corresponding equivalence class, i.e. place, is the same one you started with. | |
| Oct 30, 2012 at 11:08 | comment | added | Keenan Kidwell | $F\subseteq F^\prime$. Then set $\overline{\mathbf{Q}}_v$ equal to the directed colimit of these completions of finite subextensions. This will be an algebraic extension of $\mathbf{Q}_p$, where $p$ is the residue characteristic of $v$, containing $\overline{\mathbf{Q}}$. Therefore you can uniquely extend the usual absolute value on $\mathbf{Q}_p$ to $\overline{\mathbf{Q}}_v$. You can also choose an embedding of $\overline{\mathbf{Q}}_v$ into $\overline{\mathbf{Q}}_p$. The induced absolute value will have to coincide with the given one, and by restriction to $\overline{\mathbf{Q}}$, you get | |
| Oct 30, 2012 at 11:05 | comment | added | Keenan Kidwell | I'm not sure what uniformizer or power series you're referring to here. I can tell you how to go from a prime $v$ of $\overline{\mathbf{Q}}$ to an embedding. I'll assume $v$ is non-Archimedean. Pick an absolute value in $v$. By restriction to each finite subextension, i.e., each number field $F$, you get an absolute value, which I'll also denote by $v$, and you can form the completion $F_v$. If $F\subseteq F^\prime$, then the absolute value on the bigger field restricts to the absolute value on the smaller one, so there is a canonical injection $F_v\rightarrow F_v^\prime$ over | |
| Oct 29, 2012 at 21:42 | comment | added | Adam Hughes | Just so we're clear, the way to think of the big prime over p as an embedding is to look at a uniformizer and looking at the power series for the element of $\overline{\mathbb{Q}}$ in $\overline{\mathbb{Q}_p}$, or is there something I'm missing? | |
| Sep 13, 2012 at 5:34 | vote | accept | Adam Hughes | ||
| Sep 13, 2012 at 4:15 | history | edited | Keenan Kidwell | CC BY-SA 3.0 |
added 551 characters in body
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| Sep 13, 2012 at 4:03 | history | answered | Keenan Kidwell | CC BY-SA 3.0 |