Timeline for If $R$ is an etale extension of $\mathbb Z$, then $R = \mathbb Z^n$?
Current License: CC BY-SA 4.0
6 events
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| Jul 24, 2018 at 13:11 | comment | added | skd | @TKe I don't think the question's the problem, it was just my comment that was insufficient. The answer still works, but my comment only shows that class field theory implies that the abelianization of the etale fundamental group of Z vanishes. | |
| Jul 24, 2018 at 4:34 | comment | added | user19475 | Maybe one can edit the question accordingly? (I am not sure if I am allowed to change the question.) | |
| Jul 24, 2018 at 4:30 | comment | added | skd | @TKe Yes, you're right; sorry! I don't know anything about non-abelian class field theory, but I'd like to see a proof of this vanishing using that theory. | |
| Jul 24, 2018 at 3:53 | comment | added | user19475 | Class field theory, which can be proved algebraically, gives the vanishing of the abelianised fundamental group. Perhaps using non-abelian class field theory, one can handle the whole étale fundamental group? | |
| Jul 24, 2018 at 2:02 | comment | added | skd | By the way, the vanishing of $\pi_1^\mathrm{et}(\mathrm{Spec}(\mathbf{Z}))$ is just (by class field theory) the statement that $\mathbf{Z}$ is a PID. | |
| Jul 23, 2018 at 19:50 | history | answered | skd | CC BY-SA 4.0 |