Timeline for A mixing property for finite fields of characteristic $2$
Current License: CC BY-SA 3.0
4 events
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| Jul 24, 2012 at 19:15 | comment | added | Peter Mueller | The connection of $x^{r+1}+x-y$ with the Serre polynomial $z^{r+1}-wz+1$ is provided by $y=v^{r+1}$, $x=vz$, $v^r=w$. Note that the translates with cyclic (or the purely insepable extension given by $v^r=w$) extensions doesn't change Galois groups. | |
| Jul 24, 2012 at 18:39 | comment | added | Peter Mueller | Beautiful argument! Two minor comments: Missing detail: I think this follows from looking at the ramification of the place $y\mapsto0$ in the splitting field of $x^{r+1}+x-y$: The factorization $x^{r+1}+x=x(x+1)^r$ tells us that the intertia group fixes a point and is transitive on the remaining $r$ points. So a Sylow $2$-subgroup of $G^+$ has order $r$. But this gives $G^+=PGL(2,r)$. Reference: Cohen, Stephen D.: The distribution of polynomials over finite fields. Acta Arith. 17 1970 255–271, contains the facts used by Speyer, which are not so clear in Birch/Swinnerton-Dyer. | |
| Jul 24, 2012 at 15:18 | comment | added | Seva | Wow! I will be traveling till mid September and will hardly be able to check the details in the future foreseen, but I impressed both by the result (modulo the missing detail) and the technique involved! | |
| Jul 24, 2012 at 15:06 | history | answered | David E Speyer | CC BY-SA 3.0 |