Timeline for Irreducible polynomial over number field with roots in every completion?
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
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| Nov 30, 2009 at 0:48 | comment | added | David E Speyer | The Frobenius density theorem is a weaker version of the Cebatarov density theorem; see the second paragraph of en.wikipedia.org/wiki/… One can phrase Frobenius density without reference to a choice of generator of Gal(\ell/k), so it makes sense to ask about it when k is not finite. But, as I have just learned, it isn't true in that generality! | |
| Nov 30, 2009 at 0:43 | comment | added | David E Speyer | (Continued) So what we really have here is a failure of Frobenius density: the groups D aren't being distributed properly as \ell varies. And the reason this can happen is that you can't form the zeta function of an infinite field; so the usual proof based on the order of the zeta function at s=1 doesn't apply. | |
| Nov 30, 2009 at 0:43 | comment | added | David E Speyer | On further thought, I take that back. Let K be FC's infinite field, and L/K the S_5 extension. Let k be a residue field of O_K, and \ell a residue field of O_L lying over k. There is still a subgroup D of Gal(L/K) which surjects onto Gal(\ell/k). My complaint above was that we didn't have a specified generator for Gal(\ell/k) but that's OK -- it's a trivial group anyway! | |
| Nov 30, 2009 at 0:17 | comment | added | David E Speyer | Let k be a residue field of O_K. Because K/Q is infinite, it is possible that k is infinite. Then one can't characterize the elements of k as the fixed points of some Frobenius map. In my opinion, this is the largest break in the argument, although probably not the only one. | |
| Nov 29, 2009 at 8:05 | comment | added | Rebecca Bellovin | That's interesting, do you know where the argument breaks down if K/Q is infinite? | |
| Nov 29, 2009 at 5:19 | comment | added | Dror Speiser | As for Hensel, sure, but if I'm not mistaken an irreducible integer polynomial will satisfy Hensel's requirements for all but finitely many places. | |
| Nov 29, 2009 at 4:49 | comment | added | Dror Speiser | I had the cyclotomic thing in my mind a few hours before I posted, but then forgot it... | |
| Nov 29, 2009 at 4:47 | vote | accept | Dror Speiser | ||
| Nov 29, 2009 at 1:36 | history | answered | user631 | CC BY-SA 2.5 |