Timeline for Explicitly describable maximal unramified extension of a number field
Current License: CC BY-SA 2.5
16 events
| when toggle format | what | by | license | comment | |
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| Apr 23, 2011 at 5:34 | answer | added | Chandan Singh Dalawat | timeline score: 5 | |
| Mar 12, 2011 at 6:07 | comment | added | Chandan Singh Dalawat | A friend has kindly sent me Fröhlich's 1962 article. It does not have an example of a number field having an unramified extension of infinite degree. | |
| Mar 11, 2011 at 19:46 | answer | added | Franz Lemmermeyer | timeline score: 2 | |
| Mar 11, 2011 at 15:38 | comment | added | Tim Dokchitser | @Chandan: Thank you! Of course. I was thinking in a totally different direction (mathoverflow.net/questions/44801) and didn't realize the class field tower construction proves it. | |
| Mar 11, 2011 at 12:09 | comment | added | Chandan Singh Dalawat | In 1964 they found examples of number fields for which the class field tower does not terminate. But I suspect that examples of number fields admitting an unramified extension of infinite degree can be found in Fröhlich, On non-ramified extensions with prescribed Galois group, Mathematika,9, 133-134 (1962). I cannot read this paper because it costs £30.00. | |
| Mar 11, 2011 at 11:53 | comment | added | Chandan Singh Dalawat | Tim, two Russians (Golod-Shafarevich) did it first. There is a somewhat dated write-up by Roquette in Cassels-Fröhlich. | |
| Mar 11, 2011 at 9:43 | comment | added | Tim Dokchitser | Is it known (or conjectured?) that infinite unramified extensions of number fields exist? | |
| Mar 11, 2011 at 8:40 | vote | accept | Chandan Singh Dalawat | ||
| Mar 10, 2011 at 19:02 | comment | added | Cam McLeman | @Franz: Nope, don't think so. | |
| Mar 10, 2011 at 18:29 | comment | added | Franz Lemmermeyer | Is there an example of an infinite unramified extension of a number field (not necessarily maximal) and known Galois group? | |
| Mar 10, 2011 at 17:01 | comment | added | Julien Puydt | To complement above : more generally than an isomorphism with a "basic" group, completely determining a group can mean one has it as a product of "basic" groups, as extension ; that it has a local description ; that one knows what its normal subgroups are, etc (and that is a big caetera!) | |
| Mar 10, 2011 at 14:13 | answer | added | Cam McLeman | timeline score: 10 | |
| Mar 10, 2011 at 11:47 | comment | added | Chandan Singh Dalawat | @EE John: We say that an unknown group $G$ has been completely determined if we have shown it to be isomorphic to a known group $G'$. For example, when $L$ is the splitting field over $\mathbf{Q}$ of the polynomial $T^3-2$, the group $G=Gal(L|\mathbf{Q})$ is completely known once we show that it is isomorphic to $G'=\mathfrak{S}_3$. | |
| Mar 10, 2011 at 11:04 | comment | added | user13549 | What is the meaning of "Gal(M/K) has been completely determined?" | |
| Mar 10, 2011 at 8:22 | history | edited | Chandan Singh Dalawat | CC BY-SA 2.5 |
typo in the title
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| Mar 10, 2011 at 7:22 | history | asked | Chandan Singh Dalawat | CC BY-SA 2.5 |