Timeline for Is the absolute Galois group of the field of Laurent series in positive characteristic finitely generated?

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Jun 13, 2012 at 11:53	comment	added	Lior Bary-Soroker		I think the general case follows from the $\mathbb{A}^1$ case: If you take a curve $C$, then the decomposition group $D$ of $C$ is an open subgroup of a decomposition group $D'$ of $\mathbb{A}^1$. Hence if $D'$ is not finitely generated, then $D$ is not finitely generated.
Jun 13, 2012 at 10:47	comment	added	Filippo Alberto Edoardo		Actually you can deduce it from class field theory, without any assumption on $K$ (beyond being of characteristic $p$). You can find the argument in the paper arxiv.org/pdf/1005.2289v1 on the Remark at page 4. The argument works for global fields, but the proof actually shows that the decomposition group of localizations is big enough, which is what you want. It holds for general one-dimensional global fields of positive characteristic, not only for the curve $\mathbb{A}^1$.
Jun 13, 2012 at 9:16	history	answered	Lior Bary-Soroker	CC BY-SA 3.0