Timeline for Global class field theory and closure of unit groups
Current License: CC BY-SA 4.0
12 events
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| Nov 6, 2023 at 20:29 | comment | added | David Loeffler | This is the correct statement for (2). For (1) your formulation is correct if $K$ itself is totally real; but if $K$ isn't totally real, "the maximal totally real subfield of $K^{\mathrm{ab}}$" will not contain $K$! You want to replace "totally real" with "unramified at the real places of $K$" (i.e. as close to totally real as $K$ itself is). | |
| Nov 6, 2023 at 18:50 | history | rollback | Tim |
Rollback to Revision 5
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| Nov 6, 2023 at 18:45 | history | undeleted | Tim | ||
| Oct 26, 2023 at 18:45 | history | deleted | Tim | via Vote | |
| Oct 26, 2023 at 17:33 | comment | added | Tim | @Aurel I see. The right statement should be, for (1), the Galois group is isomorphic to the quotient of $\mathbf{A}_K^{\times}$ by the closure of $K^{\times}\cdot(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times, 0}$ (where the superscript $0$ means "connected component of the identity"), and for (2) it should be the same but with $(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times, 0}$ replaced by all of $(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}$. Am I right? | |
| Oct 26, 2023 at 16:50 | comment | added | Aurel | I don't think that 1) is true of the class group of $K$ is nontrivial, and similarly for 2) if the narrow class group is nontrivial. | |
| Oct 26, 2023 at 15:58 | history | edited | Tim | CC BY-SA 4.0 |
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| Oct 26, 2023 at 15:44 | history | edited | Tim | CC BY-SA 4.0 |
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| Oct 25, 2023 at 10:59 | history | edited | Tim |
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| Oct 24, 2023 at 20:55 | history | edited | Tim | CC BY-SA 4.0 |
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| Oct 24, 2023 at 20:27 | history | edited | Tim | CC BY-SA 4.0 |
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| Oct 24, 2023 at 20:20 | history | asked | Tim | CC BY-SA 4.0 |